But pretty much everything that used to look "singular" about black holes has completely disappeared in our modern understanding of black holes. The singularity at the event horizon is just a coordinate singularity, an artifact of coordinates. The singularity at the black hole center may be "more real" but you can't really measure its properties by apparatuses that survive for a suffficient amount of time. The black hole itself doesn't qualitatively differ from other "kinds of matter"; ...I agree with this. There seems to be a widespread belief that a new theory of quantum gravity will be needed to understand black holes. I think not.
The main message is "Please don't send me would-be deep e-mails about black holes if you still believe that the singularity is what is important in a black hole." Everyone who believes in this misconception is a 100% layman who hasn't started to understand general relativity (let alone quantum gravity) at all.
On the outside, a black hole just appears like a heavy black planet. We can only see what is outside the event horizon. The event horizon requires tricky spacetime coordinates, but there is no singularity. If you go in past the event horizon, you cannot get out.
Inside that black hole, there is a much smaller (as viewed from the outside) region with a theory horizon. If you go in past the theory horizon, then our physics theories break down, and we have no idea what happens. Maybe there is a singularity on the inside, maybe there is infinite volume on the inside, maybe there is unification of all the forces, maybe there is supersymmetry, we just don't know. And we will never know, because we cannot get past the event horizon and report what we see.
Some wise guy is probably going to ask me why I have another blog called Singular Values if I do not believe in singularities? The answer is the title does not refer to infinities.