Seeing as this is the top result on Google for "static class java" and the best answer isn't here I figured I'd add it. I'm interpreting OP's question as concerning static classes in C#, which are known as singletons in the Java world. For those unaware, in C# the "static" keyword can be applied to a class declaration which means the resulting class can never be instantiated.
Excerpt from "Effective Java - Second Edition" by Joshua Bloch (widely considered to be one of the best Java style guides available):
As of release 1.5, there is a third approach to implementing singletons. Simply make an enum type with one element:
// Enum singleton - the preferred approach public enum Elvis { INSTANCE; public void leaveTheBuilding() { ... } }
This approach is functionally equivalent to the public field approach, except that it is more concise, provides the serialization machinery for free , and provides an ironclad guarantee against multiple instantiation, even in the face of sophisticated serialization or reflection attacks. While this approach has yet to be widely adopted, a single-element enum type is the best way to implement a singleton. (emphasis author's)
Bloch, Joshua (2008-05-08). Effective Java (Java Series) (p. 18). Pearson Education.
I think the implementation and justification are pretty self explanatory.