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No, this is nothing to do with closures. A nested class has access to its outer class’s private members, including the private constructor here.
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Read my article on beforefieldinit. You may or may not want the no-op static constructor – it depends on what laziness guarantees you need. You should be aware that .NET 4 changes the actual type initialization semantics somewhat (still within the spec, but lazier than before).
Do you really need this pattern though? Are you sure you can’t get away with:
public sealed class Singleton
{
private static readonly Singleton instance = new Singleton();
public static Singleton Instance { get { return instance; } }
static Singleton() {}
private Singleton() {}
}