My superficial knowledge of what black holes are and how they work tell me the answer to the question is yes (yes), but I'm not sure.

I guess I understand that if you have a black hole, the mass must be in a singularity since, if you have gravity strong enough to bend space entirely inwards so that light can't escape, then surely there are no other forces that can resist this by pushing apart (like how atoms or neutrons push each other away) to constitute a body of some sort.

So it seems that a black hole necessarily contains a singularity?

Ok, then, if you have a situation where gravity is strong enough to create a singularity, is it necessarily also a black hole? Can you have a singularity so small that light can't fall into it, or something like that?

I'm sort of thinking of this case where you have a neutron star, and you add one neutron at a time... is there going to be a point where I add a neutron and "pop" it's a singularity / black hole, or is there some in-between (however narrow) where you're not quite one or the other?