Traditionally, a singleton is usually implemented as

public class Foo1
{
    private static final Foo1 INSTANCE = new Foo1();

    public static Foo1 getInstance(){ return INSTANCE; }

    private Foo1(){}

    public void doo(){ ... }
}

With Java's enum, we can implement a singleton as

public enum Foo2
{
    INSTANCE;

    public void doo(){ ... }
}

As awesome as the 2nd version is, are there any downsides to it?

(I gave it some thoughts and I'll answer my own question; hopefully you have better answers)