What is the use of the enum singleton in Java? [duplicate]

This question already has an answer here:

• Implementing Singleton as Enum in Java 5 answers

When the Gang of four introduced the singleton pattern, they also had to explain, why not to use static class fields and method instead. The reason was: the possibility to inherit. For Java it had sense - we cannot normally inherit the class fields and methods.

Later the "Effective Java" book appeared. And we know now that the existence of reflection destroys the singularity of the singleton class with private constructor. And the only way to make a real SINGLEton is to make it as a single item of an enumeration. Nice. I had done some myself this way.

But a question remains: While we cannot inherit from enumeration, what is the use of this singleton? Why we don't use these old good static/class fields and methods?