How to create .NET interface with static members?

While I agree with other posters that singletons are very much over used, a possible solution to your question is to provide an abstract base class with a type parameter of the derived singleton:

public abstract class Singleton<T> where T : Singleton<T>
{
  private static T _instance;

  public static T Instance
  {
    get { return _instance; }
    protected set { _instance = value; }
  }
}

Any class that derives from Singleton will have a static Instance property of the correct type:

public class MySingleton : Singleton<MySingleton>
{
    static MySingleton()
    {
        Instance = new MySingleton();
    }

    private MySingleton() { } 
}

Before using something like this though you should really think about whether a singleton is required or if you are better off with a normal static class.

An Interface can't, to my knowledge, be a Singleton since it doesn't actually exist. An Interface is a Contract that an implementation must follow. As such, the implementation can be a singleton, but the Interface can not.