Now, let yourself move away from black holes and think about an electron.
I was talking about this with my friend earlier, and I decided that the singularity must have a very small volume, rather than zero... ie a sphere with radius of one Planck length. I guess the same must be true of an electron.

That's one reason I'm always quick to point out that QM spin is not other spins.
Can it be tought of more as 4D spin? I can visualise the roatation of a hypercube, is this sort of what electrons do?

Aside from that, are you bothered by the center of a spinning rigid object?
Good question. I'll assume a perfect sphere under no other influence. What is the exact centre doing? It must be rotating because the rest of the sphere is doing so, and in a measureable way.