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Actually, the only BH that has a formalised singularity is a non rotating BH that obeys the Schwarzchild Metric. Since BH's will be rotating (because they formed from a rotating object), the actual presence of a singularity can be called into question. If defined, the singularity will not be a point but a ring shaped object because the hole is rotating. So, the physics of the "singularity" are different and the maths describing are as well. A BH with these characteristics obeys the Kerr Metric (discovered by Roy Kerr back in the early 60's). These types of BH's are a little more complicated than your Schwarzchild hole.
In any case, if you have a complete law of quantum gravity, you can dispense with the singularity altogether because at a quantum level, the energy and mass of the hole become part of spacetime/superspace.
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