Why is the natural singularity $r=0$ in Schwarzschild geometry a spacelike one? Why is the natural singularity $r=0$ in Schwarzschild geometry a spacelike one?  
 A: Nice question. Topologically, a singularity isn't a point or set of points. It's treated as a hole in the manifold. Therefore it doesn't have its own topology or geometry. We can't even say what its dimensionality is. So if we want to define what is a spacelike or timelike singularity, we need to define it in terms of the nearby spacetime, which is a point-set and does have a geometry.
A timelike singularity is one such that there exists an observer (i.e., a timelike world-line) who has it both in his past and in his future light cones.
Given that definition, I think it should be pretty clear why a black hole singularity is not timelike. It's in your future light cone, because you can fall into it. It's not in your past light cone, because we don't observe things popping out of it.
Black hole singularities can form by gravitational collapse. If timelike singularities could form by gravitational collapse, it would be shocking, because the laws of physics can't predict what could pop out of such a singularity, and therefore the laws of physics would lose their power to predict what happens in our universe.
