Java: Static Class?

Just to swim upstream, static members and classes do not participate in OO and are therefore evil. No, not evil, but seriously, I would recommend a regular class with a singleton pattern for access. This way if you need to override behavior in any cases down the road, it isn't a major retooling. OO is your friend :-)

My $.02

Private constructor and static methods on a class marked as final.

According to the great book "Effective Java":

Item 4: Enforce noninstantiability with a private constructor

- Attempting to enforce noninstantiability by making a class abstract does not work.

- A default constructor is generated only if a class contains no explicit constructors, so a class can be made noninstantiable by including a private constructor:

// Noninstantiable utility class
public class UtilityClass
{
    // Suppress default constructor for noninstantiability
    private UtilityClass() {
        throw new AssertionError();
    }
}

Because the explicit constructor is private, it is inaccessible outside of the class. The AssertionError isn’t strictly required, but it provides insurance in case the constructor is accidentally invoked from within the class. It guarantees that the class will never be instantiated under any circumstances. This idiom is mildly counterintuitive, as the constructor is provided expressly so that it cannot be invoked. It is therefore wise to include a comment, as shown above.

As a side effect, this idiom also prevents the class from being subclassed. All constructors must invoke a superclass constructor, explicitly or implicitly, and a subclass would have no accessible superclass constructor to invoke.

Sounds like you have a utility class similar to java.lang.Math.
The approach there is final class with private constructor and static methods.

But beware of what this does for testability, I recommend reading this article
Static Methods are Death to Testability