GR is a theory expressed in terms of maths. The singularities it produces are mathematical singularities.
That said, most physicists believe that the singularities are really just the mathematics throwing up its hands and saying "I dunno". That is, we believe that somewhere between the regimes we have tested and the singularity GR stops being an accurate model of reality. And we hope that a future theory of (quantum?) gravity won't include singularities.
Infinities do not play well with other numbers. In no small part because infinity in NOT a number. In most theories infinities that pop up in the math vanish when proper physical modeling is applied. In quantum mechanics and cosmology some still persist - probably because of reliance upon dubious approximations applied to their modeling. The resolution of these infinities will be a welcome relief to physicists, who have lost their limited sense of humor at being face smacked by them.