The usual explanation for what's going on inside a black hole goes something like "General Relativity predicts a singularity with infinite curvature, but when matters gets so tightly compressed we should expect quantum effects to become important, so the singularity will probably be blurred". Authors then proceed to say that we need a quantum theory of gravity to answer such things, and quickly abandon the subject.
Well, while we don't have an experimentally confirmed quantum theory of gravity, we do have a speculative one, string theory. What does it say about singularities? Are they smeared out like the books say? Do they remain points (or rings for a rotating black hole) with infinite curvature and density? Do we not know?
Best Answer
According to the fuzzball proposal in string theory, black holes are actually horizonless and regular solutions. For some systems in five dimensions made with bound states of intersecting branes this has been already proved directly in supergravity: there are solutions without horizons and singularities, with the same asymptotic charges (Mass, Angular Momentum, Electric-Magnetic Charge) of the black hole and with a behavior that mimics what you would expect from a black hole (for instance if you scatter a quantum inside this geometry, it takes a very long time for the quantum to get out). This black holes are a kind of degenerate state of matter, but there is not really an "interior": the space cap off near the would-be horizon instead of ending into a singularity. A very strong implication is that quantum gravity physics is relevant even at the would-be horizon scale. No information paradox occurs, since there are no horizons. The singularity is replaced by a smooth (but deep) cap, and the shape of the cap discriminates between different microstates. Basically it is the extended nature of the string and the fact that it is oscillating in the non compact space that save you.
The dynamics of these objects is still poorly understood, and some people (including me) are working to find regular solutions in four dimensions. Some solutions with AdS asymptotics have already been found.
The idea fits into the bigger picture of thinking of gravity as an emergent phenomenon. In particular the singular black hole is a coarse grained and average description of the underlying microstates in which you trace (you ignore) every detail inside the would-be horizon. Indeed it's only going near one of these objects that you can find difference between this picture and the naive one.
The most general and quantum black hole will probably require a full understanding of the non-perturbative picture of string theory. But still, if we find fuzzballs in four dimensions, this will at least solve the information paradox and describe some possible states for the black hole.