We are far from understanding the physics involved in black hole singularities, but what we do know is that the singularities that appear in General Relativity are a hint of incompleteness of the theory.

It is generally thought that quantum gravity, whatever that is, will introduce some minimal length, e.g. the Planck length, in a way conceptually similar to what is hbar in the phase space of quantum mechanics.

In this regime the classical notion of geometry, and therefore of curvature and singularities, would have be replaced by something different.

It is perhaps possible then to bridge with GR, working out some semiclassical models. For instance in some recent work we have shown that a semiclassical model of affine quantum gravity/cosmology can be cured of the initial cosmological singularity, avoiding the scale factor in a FLRW cosmological model to vanish. Something similar also happens in black holes.

Article Affine Quantization and the Initial Cosmological Singularity

In QM descriptions of black holes, the beastie that is expected to be sitting at the centre of a black hole is often referred to as a "fuzzball", or a "quantum fuzzball".

If the massenergy of a black hole was truly singular, then as you fell into the hole, the effective horizon between you and the singularity would look to you like a shrinking black hole, and the Hawking radiation that you'd experience would keep increasing the further you fell. As the distance between you and a supposed central singularity drops to zero, the strength of the outward Hawking radiation pressure is supposed to increase to infinity, which - depending on what other infinities are involved, and how strong they are - might suggest Hawking radiation as one possible mechanism for preventing singularity formation.

If you can be prevented from hitting the central singularity by outward Hawking radiation-pressure, then perhaps so too can everything else that has fallen into the hole, and if nothing can reach a singularity, then perhaps the singularity can't form in the first place (especially since Hawking radiation bleeds massenergy out of a black hole, and the radiation for an arbitrarily-small hole is so intense that it evaporates away the hole's massenergy in an arbitrarily-short period of time).

Putting it another way, as you fall into a black hole whose massenergy is all initially assumed to exist in the form of a central singularity, then as you fall, you encounter greater and greater levels of Hawking radiation within r=2M on the way down, and this means that the region inside r=2M appears to you to contain energy and particles that are NOT located at the singularity point. As you fall deeper, you encounter a stronger and stronger flux of massenergy inside 2M but outside the singularity, and as you finally reach the singularity position, then, assuming that there's enough energy left for there to be a significant object at that location (and that you haven't already passed most or all of the hole's massenergy on your way down), you should see the remaining tiny "effective horizon" separating you and whatever's left at r=0 to be radiating so intensely that you see it radiate away all its remaining massenergy into the surrounding area as a final burst of Hawking radiation before you hit it.

And this is an idealised best-case scenario that ignores things like the fact that you can't "free-fall" to r=0 if you're being opposed by a radiation pressure and Hawking particle density that gets progressively higher towards the region's centre. As these things resist your freefall, you feel stronger and stronger physical acceleration effects, and as a physically accelerated (non-freefall) observer you then also see an increasingly strong Unruh radiation component.

SO:

If we start out by trying to give the "singularity" idea the benefit of the doubt, and then take into account quantum effects, then the observable physics inside r=2M would seem to correspond with what we would expect if no such singularity existed. Instead of getting a description of an empty central region containing a central sharply-defined point-mass, the core should appear to a local infalling explorer to be a hot region filled with radiation and material, whose density steady increases towards the centre of the region, with nothing singular to see at the centre. Also, given that the central region should contain so much of the hole's massenergy spread out (by radiation), there'd not even be any particular reason to assume the "pointy" gravitational tidal-forces "footprint" of a central singularity - so not only would an observer outside r=2M not be able to verify the singularity's existence, it might be that there is no obvious primary or secondary evidence to support the existence of a singularity, even for hypothetical observers //inside// r=2M, even if they were right by r=0, or right at r=0.

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Note that his argument goes well beyond the usual "censorship hypothesis" that singularities could exist but always be cloaked from direct observation by observational horizons. In //this// argument, observers inside r=2M, whose experiences define the idea of a black hole's internal physics, see a distribution of massenergy inside r=2M that doesn't agree with the idea that the hole's massenergy is all concentrated at a central point. They don't just fail to see evidence for the singularity, they see what appears to them to be counterevidence.