Thanks for this comment, it was enlightening. I'd never much thought about this because physics is really not my forte, but after a few minutes of thinking about it, it's really intuitive why a point singularity can't spin. I can't explain it in any properly rigorous or technical way, but it makes no sense that someone could (by convention, of course) designate a "front" and "rear" to a mass with zero length in every direction.