We do know that black holes can and sometimes do have angular momentum, as described by the Kerr metric. Though I have not found anything about the description of the angular momentum of the contained singularity.
Since a point cannot support rotation or angular momentum in classical physics (general relativity being a classical theory), the minimal shape of the singularity that can support these properties is instead a ring with zero thickness but non-zero radius, and this is referred to as a ringularity or Kerr singularity.
The question I believe is not easily answered, because singularities are said to have no spatial extent, and so should not possess classical angular momentum, but rather quantum spin (that elementary particles have because they do not have any spatial extent either).
In our mathematical model of particles, the intrinsic spin of a fundamental particle behaves exactly like that centre point.
The only thing I found was the ringularity, but that still has zero thickness. In many cases on this site, singularities are compared to elementary particles (both being pointlike, zero dimensional), so the question comes up whether black holes' singularities can have classical or quantum spin.
Electrons - and all other elementary particles - may be viewed as microstates of very tiny black holes.
As far as I understand, there are many similarities between singularities and elementary particles, so their spin might be related too.
- Can singularities' spin be related to the quantum spin of elementary particles?