Based on the definition supplied, a singularity could be anything from a point mass to a mass whose physical dimensions are just less than its associated event horizon. Both of these are likely extremes, with the true character and description residing somewhere in between. Still in either case, any credible description of a black hole, would also represent a description of a gravitational singularity, consistent with our current understanding and observations.
I don't think that is consistent with QM. And forgive my impertinence, but aren't (theoretical) singularities dimensionless? If so, then any object that occupies volume is not a singularity by definition.
I am not sure what is or is not consistent with QM. Certainly attempts to develop a model of quantum gravity have not yet been successful. Even GR is having some difficulties keeping up with our ever expanding observations at macrocosmic scales, I.e. the need for dark matter and dark energy, which remain an unknown.
But what was being discussed was the black hole and singularities. At least it was that portion of the discussion I was initially commenting on... And again anything that either GR or QM has to say about what lies within the event horizon remains theoretical.., in the extreme. Even much of what occurs at and just outside the event horizon, being beyond direct observation and/or experiment, must of necessity remain the subject of theory.
My earlier point was that singularity, within the context of this thread was poorly defined. Yes if one clearly states that the definition to be accepted here is confined to that of the theoretical application of GR, one could assume that a singularity is a point mass.., having no dimensions. But that is not the only definition even in physics of a singularity. There can easily be a difference between a singularity and a point singularity.
A singularity, within the context of physics, could be defined as a region of space or space-time where mass has reached an absolute or maximum density, rather than an infinite density. Two things follow from this. First, a region of space would not normally be thought of as a point, rather than a volume, and a singularity would then have some dimension. And second, substituting absolute or maximum for infinite, again allows that the singularity involved might have dimensions rather than exist as a dimensionless point. All we can really say with any degree of certainty supported by observation, is that whatever lies within the event horizon seems to prevent the escape even of light.
A popularized view of this from the standpoint of GR, is that the escape velocity within the even horizon is greater than the speed of light, so no light escapes. If light cannot escape, then nothing does... However (and I offer the following only as a thought experiment, not as an alternate model), it might be just as reasonable to assume that the conditions within the event horizon are such that photons are never generated.., that matter as we understand it does not exist as independent particles, in which case the escape velocity, or even effective gravitational force, would be no issue, as there would be nothing to escape. In either case what we see from the outside would be the same.., nothing including light escaping the black hole from inside the event horizon.
A long winded way around saying that the word singularity in physics, does not automatically mean "point" singularity, and it may in fact refer to a region of space/space-time which has some volume.