What is spinning in a spinning black hole? When a rotating star collapses into a neutron star, the resulting object spins at a huge number of rpm due to its much smaller volume and the conservation of angular momentum. What happens when a rotating star collapses into a black hole? Presumably, we get a spinning black hole, but if the star has collapsed into a point with zero volume, what is it that is spinning?
 A: Kerr black holes (i.e., spinning black holes, and also the particular case of a non-spinning black hole) are vacuum solutions of General Relativity. In other words, they are derived by assuming there is no matter anywhere in spacetime. While people often interpret that all of the mass, for example, is concentrated on the singularity, in classical General Relativity the singularity is not a part of spacetime. It is actually where spacetime ends. As mentioned in the comments to the question, it is also worth recalling that the singularity in a spinning black hole is not pointlike.
Hence, there is this really interesting property in black holes in which they are made of nothing. Yet, they have mass and angular momentum. One common interpretation for this is that this is the mass and angular momentum carried by the gravitational field itself. In the same way that (for example) the electromagnetic field can carry energy and angular momentum, so can the gravitational field. Recall that in General Relativity "mass" and "energy" are essentially the same concept.
I find this question somehow similar to a famous angular momentum paradox by Feynman. In field theory, one can often have angular momentum laying around and not notice anything rotating.
At last, General Relativity isn't particularly trustworthy near the singularity. We expect that quantum effects will be really relevant at that scale, and hence there might not even be a singularity at all. Hence, there is a chance quantum effects leave a remnant behind to spin. Nevertheless, do notice that no one currently knows quantum gravity in a reliable way, and hence we can't really tell what goes on near the singularity.
