I think you're mixing two different things here. The singleton pattern calls for a single instance that is used by all callers. Inheritance just means I can share common logic between a class hierarchy. This, I feel, is an implementation of the singleton pattern:
(ignore the lack of locking/thread safety, for the example's sake)
public class Singleton
{
private static Singleton _instance;
public static Singleton Instance
{
get
{
if (_instance == null)
{
// This is the original code.
//_instance = new Singleton();
// This is newer code, after I extended Singleton with
// a better implementation.
_instance = new BetterSingleton();
}
return _instance;
}
}
public virtual void ActualMethod() { // whatever }
}
public class BetterSingleton : Singleton
{
public override void ActualMethod() { // newer implementation }
}
We still have a singleton, accessed through the Singleton class's static Instance member. But the exact identity of that instance can be extended through subclassing.