I think I've seen it suggested: a hyper-dense object of some kind, which isn't a singularity.

Maybe some other effect is in play that we're ignorant to, which prevents singularity formation. This might be a clean way of explaining away why our current math breaks down at the center of these things.

To be clear, I'm not sold on the idea myself. But, if these objects are singularities (I believe this to be the case), the strangeness of them is massively underrated.