What goes on inside a black hole's singularity?

It would certainly seem that, inside the Schwarzschild radius of a black hole, our understanding of physics and quantum mechanical behavior of particles and radiation breaks down. Most physicists and cosmologists are confident enough of that seemingly self-evident fact to state it as fact. But do you think that means that the elementary particles of the standard model; the leptons, quarks, and bosons; are just packed together so tightly that they have no wave function and all physics disappears? Does it mean that they still exist and act, only under different rules? Or perhaps they return to some primordial state in which there are no particles and there is no standard model.

How could we know the singularity has zero volume and infinite density? It would seem that could only be true if all particles are point particles with no real volume. And if that’s the case, then not only is non-locality an illusion as suggested by quantum entanglement, the very existence of “stuff” is an illusion.

I’m not reaching for a bunch of woo-woo here, but if quantum mechanics has taught me anything it is that I can’t understand all of physics simply based on what illusions I see at the limited wavelength and macro scale of my incredibly narrow viewport. Dawkins described human observational limits aptly as our being clothed in a burka with a 1-inch slit to look out of, but being unaware that the fabric of that burka towers hundreds of miles above and hangs hundreds of miles below that viewing slit. We have eyes designed to take in just a tiny fraction somewhere in the middle of a vast spectrum of radiation. We cannot take in the scale of the solar system, and it is just a tiny dot in our galaxy, which is a tiny dot in our local cluster, which is a tiny dot in a known universe with a current diameter of 94 billion light years. We have eyes that can see a rock, but cannot see that it is almost entirely empty space and cannot even imagine the weirdness of the quantum world that makes up the Earth we walk on, the bodies that we inhabit, and the brains we use to think about these things.

Yet amazingly, theists ask, “Without god, where would you get the sense of wonder, the appreciation of majesty?”

I know that we do not currently know the answers. I’m just looking for your best guess here. What goes on inside a black hole’s singularity?