It might be difficult to define singular semi-Riemannian manifolds applied to general relativity and that might be the reason why in quantum field theory and string theory topological smooth manifolds seem to suffice. The idea might be that in cosmologcial terms every singularity gets smoothened no matter how bad it is whether caught in a big bang or what naught.
There are of course non-smooth manifolds with degenerate metrics such as in all kinds of black holes.
So, as far as general relativity is concerned it appears that the idea is both useful and state of the art alright.
