A Singleton isn't just a static class that can't be inherited. It's a regular class that can be instantiated only once, with everybody sharing that single instance (and making it thread safe is even more work).
The typical .NET code for a Singleton looks something like the following. This is a quick example, and not by any means the best implementation or thread-safe code:
public sealed class Singleton
{
Singleton _instance = null;
public Singleton Instance
{
get
{
if(_instance == null)
_instance = new Singleton();
return _instance;
}
}
// Default private constructor so only we can instanctiate
private Singleton() { }
// Default private static constructor
private static Singleton() { }
}
If you're going to go down the path you're thinking, a static sealed class will work just fine.