Singularities exist in theoretical 'perfect' solutions to General Relativity, but when you look at actual natural Kerr-like objects spinning in a noise filled background of GR waves and other incoming radiation and matter, its likely that no physical real singularities exist.
Brandon Carter, referring to spinning black holes (all real black holes spin):
“Thus we reach the conclusion that a timeline or null geodesic
or orbit cannot reach the singularity under any circumstances
except in the case where it is confined to the equator, cos() = 0
…..Thus as symmetry is progressively reduced, starting from the
Schwarchild solution, the extent of the class of geodesics reaching
the singularity is steadily reduced likewise, … which suggests that
after further reduction in symmetry, incomplete geodesics
may cease to exist altogether”
Kerr Fields, Brandon Carter 1968. http://luth2.obspm.fr/~luthier/carter/trav/Carter68.pdf
Thus we reach the conclusion that a timeline or null geodesic or orbit cannot reach the singularity under any circumstances except in the case where it is confined to the equator, cos() = 0 .... Thus as symmetry is progressively reduced, starting from the Schwarchild solution, the extent of the class of geodesics reaching the singularity is steadily reduced likewise, … which suggests that after further reduction in symmetry, incomplete geodesics may cease to exist altogether
Kerr Fields, Brandon Carter 1968. (NB: PDF)
No hair theorems don't make singularities any more likely as they speak to the ring - down time of a black hole in a perfectly quiet background. Its the natural stochastic infall of stuff that keeps real singularities from forming.
