What is the best approach for using an Enum as a singleton in Java?

Suppose you're binding to something which will use the properties of any object it's given - you can pass Elvis.INSTANCE very easily, but you can't pass Elvis.class and expect it to find the property (unless it's deliberately coded to find static properties of classes).

Basically you only use the singleton pattern when you want an instance. If static methods work okay for you, then just use those and don't bother with the enum.

A great advantage is when your singleton must implements an interface. Following your example:

public enum Elvis implements HasAge {
    INSTANCE;
    private int age;

    @Override
    public int getAge() {
        return age;
    }
}

With:

public interface HasAge {
    public int getAge();
}

It can't be done with statics...

(Stateful) Singletons are generally used to pretend not to be using static variables. If you don't actually use the publicly static variable then you will fool less people.

I would choose the option which is the simplest and clearest. This is somewhat subjective, but if you don't know what is clearest, just go for the shortest option.