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Is it mandatory utility class should be final and private constructor?

By making private constructor, we can avoid instantiating class from anywhere outside. and by making class final, no other class can extend it. Why is it necessary for Util class to have private constructor and final class ?

This is not a mandate from a functional point of view or java complication or runtime. However, it's a coding standard accepted by the wider community. Even most static code review tools, like checkstyle, check that such classes have this convention followed.
Why this convention is followed is already explained in other answers and even OP covered that, but I'd like to explain it a little further.
Mostly utility classes are a collection of methods/functions which are independent of an object instance. Those are kind of like aggregate functions as they depend only on parameters for return values and are not associated with class variables of the utility class. So, these functions/methods are mostly kept static. As a result, utility classes are, ideally, classes with only static methods. Therefore, any programmer calling these methods doesn't need to instantiate the class. However, some robo-coders (maybe with less experience or interest) will tend to create the object as they believe they need to before calling its method. To avoid that, we have 3 options:
Keep educating people to not instantiate it. (No sane person can keep doing it.)
Mark the utility class as abstract: Now robo-coders will not create the object. However, reviewers and the wider java community will argue that marking the class as abstract means you want someone to extend it. So, this is also not a good option.
Private constructor: Not protected because it'll allow a child class to instantiate the object.
Now, if someone wants to add a new method for some functionality to the utility class, they don't need to extend it: they can add a new method as each method is independent and has no chance of breaking other functionalities. So, no need to override it. Also, you are not going to instantiate it, so no need to subclass it. Better to mark it final.
In summary, instantiating a utility class (new MyUtilityClass()) does not make sense. Hence the constructors should be private. And you never want to override or extend it, so mark it final.

It's not necessary, but it is convenient. A utility class is just a namespace holder of related functions and is not meant to be instantiated or subclassed. So preventing instantiation and extension sends a correct message to the user of the class.

There is an important distinction between the Java Language, and the Java Runtime.
When the java class is compiled to bytecode, there is no concept of access restriction, public, package, protected, private are equivalent. It is always possible via reflection or bytecode manipulation to invoke the private constructor, so the jvm cannot rely on that ability.
final on the other hand, is something that persists through to the bytecode, and the guarantees it provides can be used by javac to generate more efficient bytecode, and by the jvm to generate more efficient machine instructions.
Most of the optimisations this enabled are no longer relevant, as the jvm now applies the same optimisations to all classes that are monomorphic at runtime—and these were always the most important.

By default this kind of class normally is used to aggregate functions who do different this, in that case we didn't need to create a new object

Java Class That Has 90% Static Members. Good Practice Or What?

I'm 14 and have been learning java for about 4/5 months. I'm coding a game now called super mario winshine and i wanted to know if it is good practice to have a class that is mostly static variables.
The class is the one that holds all the information for the game's level/world. Since I only need one version of this class, and lots of other classes will be using it, I choose to make all the variables static. Is this good practice?
I have considered the fact that i could keep the variables "non-static" and just clone the main object i use for that class, but I thought i would rather sacrifice "O-O" for memory in this case.

As soon as you want to have two or more worlds this will fail. Say, when your first release is a runaway success and you want to add the "parallel universe" expansion set.
In my experience, 90% of the time when marketing says "oh, don't worry, there will only be one Application/Window/Database/User" they are wrong.
ADDED
I would also avoid using a true Singleton pattern with World.getInstance() etc. Those are for the rare cases where it really is an essential requirement that there only be one of something. In your case, you are using it as a convenience, not a requirement.
There is no perfect fix, YMMV, but I'd consider a single static method, something like
World World.getWorld(String name)
and then you call real (non-static) methods on the World that is returned. For V1 of your program, allow null to mean "the default world".
Some might put that method into a class named WorldManager, or, perhaps showing my age, a more clever name like Amber. :-)

It all depends upon what your methods and classes are. There is no problem in defining utility methods as static methods in a class. There is no need to make it a singleton as others are suggesting. Look at the Math class from java.lang package. It has lot of utility methods but it isn't a singleton.
Also check out static imports functionality. Using this you doesn't need to qualify method calls with the class name.

Well, what you are doing is definitely an option. Or you could use a singleton pattern:
public class World {
private static World instance = new World();
private World() {
}
public static World getInstance() {
return instance;
}
}
Now just use World.getInstance() everywhere to have a unique object of this type per application.

I would say it's definitely not a good practice.
I've not seen your code, but having several static variables in a class that other classes access freely seems to indicate that you're not really using object orientation/classes but more just writing procedural code in Java. Classes should generally encapsulate/hide all their variables - static or not - from access from other classes so that other classes don't depend on how the class is implemented.
The static part also causes problems with making threads work (global variables are hard to lock in a good way so that nothing deadlocks) and with unit testing (mocking is all but impossible)
I also agree with the other posters, if you need "global variables", at least make them singletons. That allows you to change strategy easier later and does not lock you to one world.
Edit: I'm definitely not advocating singletons as a good pattern here if someone read it like that, but it does solve some problems with static variables, esp. regarding testing/mocking compared to just statics so I'd say it's a ever so slightly lighter shade of gray :) It is also a pattern that is easier to gradually replace with better patterns by for example using a IoC container.

I think it is fine as long as you don't need anything more sophisticated, in other words, static fields are OK as long as different objects (including subclasses if there will be any) do not need different values.
You code by yourself, refactoring is easy with modern tools, me says don't fix it until it is broken, and focus on the algorithmic aspects of your project.
Perhaps you may think to encapsulate all those static fields within a different static class, as it is a good principle to "keep what changes seperate from what does not". Chances are one day you will want to initiate that static class with different values, for example want to read the initial values from an XML file and/or registry, add behaviour, etc. so instead of a static class you will implement it with a Singleton pattern.
But clearly that is not the concern of today. Till then, enjoy!

You may wish to look into implementing this class as a singleton, while there is nothing particularly wrong with your approach it may lead to some inflexibility further down the road.
Also you should take in to consideration the purpose of static members which is to be a member of the class and 'act' on/with the class not an instance of it. For example the static method in a singleton returns either a new instance of the class if one doesn't already exist or returns the instance, and because the method is static you do not instantiate a new one. This is probably worth a read because it can be somewhat confusing when determining the appropriate use of static members

I'm not sure what you are really talking about from your short description, so I'll try this:
public class Level {
static List<Mushroom> mushrooms;
static List<Coin> coins;
...
}
Is that you were describing?
You asked if this is "good practice" and I can tell you that this looks very odd, so, no, it's not.
You gain absolutely nothing by doing this. You make it impossible to have more than one Level, which brings no advantage, but it could cause trouble in the future.
I didn't understand your last paragraph where you say you made most things static to save memory. You would usually create one Level and it would be passed around (without cloning) to the various classes/methods that read from it or modify it.

Single instance with methods in java

I am wondering about programming decision - which I think is matter of style.
I need to have single instance of class which has only methods and no attributes.
To obtain that in java I have two options:
create an abstract class with static methods within, thus it will not be possible to create any instance of the class and that is fine,
use a singleton pattern with public methods.
I tend to go for second approach although met with 1. Which and why is better of those, or there is third option.

Would it make sense for that singleton to implement an interface, allowing you to mock out those methods for test purposes?
I know it goes against testing dogma these days, but in certain situations I think a static method is fine. If it's the kind of behaviour which you're never going to want to fake for test purposes, and which is never going to be polymorphic with other implementations, I don't see much point in making a singleton. (Singletons are also generally the enemy of testability, although if you only directly refer to them in the injection part of your code, they can implement appropriate interfaces so their singletoneity never becomes a problem.)
It's worth mentioning that C# has "static classes" for this kind of situation - not only do they prohibit other code from deriving from or instantiating the class, but you can't even use it as a parameter. Basically it signals the intent very clearly.
I would definitely suggest at least having a private constructor to prevent instantiation by the outside world.

My personal view is that the class should contain a private constructor and NOT be abstract. Abstract suggest to a reader that there is a concrete version of the class somewhere, and they may waste time searching for it. I would also make sure you comment your code effectively.
public class myClass {
/** This class should never be instantiated. */
private myClass() {
}
public static void myMethod() {
}
...
//etc
...
}

For option #1, it may not even be that important to restrict instantiation of your static utility class. Since all it has is static methods and no state, there is no point - but neither harm - instantiating it. Similarly, static methods can't be overridden so it does not make sense - nor difference - if it is subclassed.
If it had any state, though - or if there is a chance that it will get stateful one day - it may be better to implement it as a normal class. Still I would prefer not to use it as a Singleton, rather to pass its sole instance around via dependency injection. This makes unit testing so much easier in the long run.

If it holds a state I would use the singleton pattern with private constructors so you can only instantiate from within the class. If it does not hold a state, like the apache commons utility classes, I would use the static methods.

I've never seen the problem with static methods. You can think of static methods as somehow breaking OO, but they make perfect sense if you think of static as a marker that something is stateless. You find this in the java apis in places like java.Math. If you're worried about subclassing you can always make it final.
There is a danger in that a class like that can end up as a "utility method garbage can", but as long as the functionality doesn't diverge too much then there's nothing wrong with it.
It's also clearer, as there's no need to manage an object lifecycle like you would with a singleton (and since there's no state, what's the point of that anyway?).

For a single instance, I suggest you have an enum, with one instance.
However, for a class with no attributes, you don't have to have an instance. You can use a utility class. You can use an enum, with no instances and only static methods. Note: this cannot be easily mocked out.
You can still implement an interface if you ever need to mock out the implementation in testing.

Is it wrong to have static and non-static methods in the same class?

Is it wrong to have static and non-static methods in the same class?

Not really in regular java programming.
But if you're working extensively with dependency injection you probably have few or no static methods at all. In such a context it's fairly common to have only a few utility classes with static methods, and no other static methods.

No, it's not wrong. For example, a common use is to have static factory methods in a class definition.

I think it's fine to create static utils classes, especially when you're not really sure (yet) what the design should be, because you're still learning about the problem domain.
Staticness is a "yet to designed properly" marker. Often the static solution is perfectly adequate; but sometimes, as the project progresses, you find you do have to rewrite that "whole part", but you have (at that later stage) a far more complete understanding of the problem domain, and are therefore in a position to actually design a "proper solution" to those problems.
I think us programmers hammer ourselves unfairly about "rework". You need to do the work in order to understand the work well enough to do the work properly. I see no way past this catch 22;
I can cite many examples of static from the core API. java.lang.Math, java.util.Arrays, java.util.Collections. BUT please note that these classes are "utils classes" which exist only to provide a bunch of static methods. IMHO, The presence of static methods in a "stateful object" is just begging to be refactored.
I'll betcha that todays API designers would love to be able to split-down Integer (and the other wrapper classes)... BUT they're well and truly stuck with what they've got. Which is a warning in itself... that static implies final, and there's a darn good reason that (unlike C++) java methods can be overridden by default. Static is inherently more "binding" than non-static... down the trick you CAN NOT adapt implementations to different situations, contexts, etc, etc, etc.
Cheers. Keith.

I believe its a bad idea to have static and non-static methods in the same class with the same name. This can be very confusing. I would suggest trying to make the names different in this case.
Similarly, don't make methods with the same name but different case, or methods with the same name as the class in the same class. Both are legal, and even occur in the JDK, but are confusing IHMO.

I guess not especially when dealing with singletons.

Is there a rule of thumb for when to code a static method vs an instance method?

I'm learning Java (and OOP) and although it might irrelevant for where I'm at right now, I was wondering if SO could share some common pitfalls or good design practices.

One important thing to remember is that static methods cannot be overridden by a subclass. References to a static method in your code essentially tie it to that implementation. When using instance methods, behavior can be varied based on the type of the instance. You can take advantage of polymorphism. Static methods are more suited to utilitarian types of operations where the behavior is set in stone. Things like base 64 encoding or calculating a checksum for instance.

I don't think any of the answers get to the heart of the OO reason of when to choose one or the other. Sure, use an instance method when you need to deal with instance members, but you could make all of your members public and then code a static method that takes in an instance of the class as an argument. Hello C.
You need to think about the messages the object you are designing responds to. Those will always be your instance methods. If you think about your objects this way, you'll almost never have static methods. Static members are ok in certain circumstances.
Notable exceptions that come to mind are the Factory Method and Singleton (use sparingly) patterns. Exercise caution when you are tempted to write a "helper" class, for from there, it is a slippery slope into procedural programming.

If the implementation of a method can be expressed completely in terms of the public interface (without downcasting) of your class, then it may be a good candidate for a static "utility" method. This allows you to maintain a minimal interface while still providing the convenience methods that clients of the code may use a lot. As Scott Meyers explains, this approach encourages encapsulation by minimizing the amount of code impacted by a change to the internal implementation of a class. Here's another interesting article by Herb Sutter picking apart std::basic_string deciding what methods should be members and what shouldn't.
In a language like Java or C++, I'll admit that the static methods make the code less elegant so there's still a tradeoff. In C#, extension methods can give you the best of both worlds.
If the operation will need to be overridden by a sub-class for some reason, then of course it must be an instance method in which case you'll need to think about all the factors that go into designing a class for inheritance.

My rule of thumb is: if the method performs anything related to a specific instance of a class, regardless of whether it needs to use class instance variables. If you can consider a situation where you might need to use a certain method without necessarily referring to an instance of the class, then the method should definitely be static (class). If this method also happens to need to make use of instance variables in certain cases, then it is probably best to create a separate instance method that calls the static method and passes the instance variables. Performance-wise I believe there is negligible difference (at least in .NET, though I would imagine it would be very similar for Java).

If you keep state ( a value ) of an object and the method is used to access, or modify the state then you should use an instance method.
Even if the method does not alter the state ( an utility function ) I would recommend you to use an instance method. Mostly because this way you can have a subclass that perform a different action.
For the rest you could use an static method.
:)

This thread looks relevant: Method can be made static, but should it? The difference's between C# and Java won't impact its relevance (I think).

Your default choice should be an instance method.

If it uses an instance variable it must be an instance method.
If not, it's up to you, but if you find yourself with a lot of static methods and/or static non-final variables, you probably want to extract all the static stuff into a new class instance. (A bunch of static methods and members is a singleton, but a really annoying one, having a real singleton object would be better--a regular object that there happens to be one of, the best!).

Basically, the rule of thumb is if it uses any data specific to the object, instance. So Math.max is static but BigInteger.bitCount() is instance. It obviously gets more complicated as your domain model does, and there are border-line cases, but the general idea is simple.

I would use an instance method by default. The advantage is that behavior can be overridden in a subclass or if you are coding against interfaces, an alternative implementation of the collaborator can be used. This is really useful for flexibility in testing code.
Static references are baked into your implementation and can't change. I find static useful for short utility methods. If the contents of your static method are very large, you may want to think about breaking responsibility into one or more separate objects and letting those collaborate with the client code as object instances.

IMHO, if you can make it a static method (without having to change it structure) then make it a static method. It is faster, and simpler.
If you know you will want to override the method, I suggest you write a unit test where you actually do this and so it is no longer appropriate to make it static. If that sounds like too much hard work, then don't make it an instance method.
Generally, You shouldn't add functionality as soon as you imagine a use one day (that way madness lies), you should only add functionality you know you actually need.
For a longer explanation...
http://en.wikipedia.org/wiki/You_Ain%27t_Gonna_Need_It
http://c2.com/xp/YouArentGonnaNeedIt.html

the issue with static methods is that you are breaking one of the core Object Oriented principles as you are coupled to an implementation. You want to support the open close principle and have your class implement an interface that describes the dependency (in a behavioral abstract sense) and then have your classes depend on that innterface. Much easier to extend after that point going forward . ..

My static methods are always one of the following:
Private "helper" methods that evaluate a formula useful only to that class.
Factory methods (Foo.getInstance() etc.)
In a "utility" class that is final, has a private constructor and contains nothing other than public static methods (e.g. com.google.common.collect.Maps)
I will not make a method static just because it does not refer to any instance variables.