Just because our calculations from an incomplete theory that we know is not a full description of the universe, goes to infinity at a point, doesn't mean that real stuff does the same. But I'd rather interpret the singularity as a breakdown of our understanding - it's as far as these mathematics can take us. Divide by zero is not calculable. It has no meaning.
I'd be willing to be bet that a real 'singularity' is in a new phase of matter or more probably a new phase of space-time, or whatever fundamental 'thing' that makes up the building blocks of the universe, that allows the black hole object to exist in some stasis that is not infinite. Until Hawking radiation, as
@Laughingbuddha mentioned, evaporates it.
However I can't see a good way of actually experimentally figuring out what lies past the event horizon, as I can't see a way of getting results back 'out' back past that screen. So how could I ever know?
Perhaps if we came up with a better theory of everything we might have a different idea.
There are, of course, other ideas floating about, about black holes and what they contain. I recently read up a bit on Gravastars - essentially black holes - a few days back, where matter is compressed so much that it gets to the scale of 'superfluid space-time'. At that point the mass at a deep level is instantaneously converted to energy and space-time there goes to a different phase of being because of the influx of energy (a bit like the phase change when ice turns into water.) This new space-time exerts a negative pressure, counteracting the postive pressure of the gravitational mass. Further infalling matter from the shell of matter, after the initial set up, is converted to energy in the new space-time...which when expressed will, through the equivalence of mass and energy, produce some matter in the new space-time. Furthermore at the boundry of the superfluid vortices form, vortices that can guide and 'trap' the masses being produced into random densities.
So you have an expansionary space-time (a.k.a there is your 'dark energy') with mass and energy set in a random density distribution....sounds a lot like a new universe!
Anyway, pinch of salt time - this theory is really only being pushed by a couple of people, so it's not 'mainstream'. Is space-time a superfluid? Could it exhibit phase-changes? Is dark energy actually real anyway, as a number of results have shed some doubt on it? If the universe inside a gravastar is a superfluid it should be rotating - as it has to conserve the angular momentum of the initial star, so is our universe rotating? Are we inside a Gravastar?
Loads of questions, I'm afraid, not so many answers!