Is there a name meaning "not a singleton"?

Yes, there is a Multiton pattern, but it means something very specific. It's not simply everything that's not a Singleton.

Castle Windsor uses the term "transient" to describe all non-Singleton objects.

I personally prefer the term "non-Singleton" though.

Prototype. It is used as a scope in Spring framework to identify dependency which will always be new instance when injected.

Multi-Instance ?

http://elegantcode.com/2008/04/17/the-opposite-of-a-singleton/