What, if any, are the similarities between a geometrodynamic singularity and that of a collapsed quantum?
The word quantum is not an object (or noun if you prefer). It is an adjective or qualifier. In quantum mechanics (QM) the wave function collapses. The wave function is a probability distribution, not really a wave at all. Collapsing just means that one of the possible outcomes got selected. I imagine that by geometrodynamic singularity you mean Black Hole. In this case it is a physical object.
Thank you, No Time. I disagree that "quantum" cannot be a noun. See most online dictionaries, e.g., Answers.com: "Physics. 1. The smallest amount of a physical quantity that can exist independently, especially a discrete quantity of electromagnetic radiation. 2. This amount of energy regarded as a unit." More importantly, though, does there exist a wavefunction for a black hole singularity (Hawking?)
Yes, you could define it as a concept. However, there is no physical object/value called a quantum. The only discrete value of electromagnetic radiation is the photon. AFAIK photons do not have a minimum energy. You can come arbitrarily close to 0. There is however, a minimum length, the plank length. This implies that photons can have a maximum energy. Hawking defined a minimum black hole, a quantum black hole. However, wavefunctions deal with interactions and exist due to the Hiesenberg uncertainty principle. There would be a wavefunction defining its location or stability for instance.
If black holes are restricted as singularities in spacetime, then how can one determine their momentum's range (according to Heisenberg's uncertainty principle)? I guess you are saying that they can still be singularities yet possess a spatial uncertainty.
I don't know what you mean by restricted as singularities. They are singularities. Mostly that means that you can not know what is in the singularity region. The only thing you can know is effects outside the singularity region. All known black holes are large massive objects. Quantum rules like wavefunction just don't apply. Heisenberg's uncertainty principle says you can know momentum or spatial location. Not both. Hawking's quantum black holes have never been observed. I've seen some speculation that electrons are quantum black holes. If there is any meaning to this at all then reality would be quite distinct from Hawking's ideas.
Don't forget that black holes come about from General Relativity, not Quantum Theory. The two are still fundamentally incompatible, and inconsistencies arise in regimes where both must be taken into account, e.g. at the Planck scale. That's why we need a theory of Quantum Gravity, which I'm guessing is what you'd need to get a good answer to your question.
belliott4488 tells of no answers without a new theory; NoTime tells of a new theory without answers at present. Pardon my skepticism; the fault lies in the physics, not you all. Hawking's paper (Phys Rev D, 15 Jan 1976, p. 191-197) is one of my favorites: Black Holes and Thermodynamics, where he first proposed what would later be known as Hawking radiation. It may be as close as we have come to a theory of quantum gravity. A radiating (without absorbing) black hole emits quanta where its end product, I would guess, is a quantum black hole.