Astronomy

Depends what you mean by "our current theories". In classical General Relatively the answer is pretty conclusively no but many people think that a quantum theory of gravity should be able to remove the singularities. In fact, this article is about an attempt to do just that with a fairly natural extension to GR (albeit one that is only mathematically tractable in 5 or more dimensions) and seems to have succeeded for the static spherically symmetric case at least.