Singleton pattern with combination of lazy loading and thread safety

You first design is actually lazy. Think about it, the instance is only created when the class is initialized; the class is only initialized when the getSingleton() method is called [1]. So the instance is only created when it's asked for, i.e. it's lazily created.

[1] http://docs.oracle.com/javase/specs/jls/se7/html/jls-12.html#jls-12.4.1

The best way is actually to use the Enum Way:

public enum Singleton {
    INSTANCE;
    public void execute (String arg) {
            //... perform operation here ...
    }
}

Your second code snippet is, in my opinion, the best way of thread-safe lazily initializing a singleton. It actually has a pattern name

Initialization-on-demand holder idiom

I would suggest you use it.

The second one is very bad in terms of readability, first one is suitable. Have a look at this article. Its about double check locking, but also will give you wide information about singletons multithreading.