I don't know what happened to my previous answer but I'll try again.

Scientists are pretty sure that black holes exist because the astronomical evidence is pretty conclusive. "Know" might be too strong of a word - a strong inference would probably be better.

The singularity is another matter. All the known laws of general relativity and quantum mechanics say that a singularity is inevitable once neutron degeneracy is overcome. However GR and QM can't be reconciled and until a theory that unites them is developed, such as string theory or quantum gravity, no one can know for sure.