Why are we so sure that there is a singularity inside the event horizon of a black hole? The Schwarzchild solution is derived in vacuum, and we find that light cones always move inward when r < 2M. So if a spherically symmetric, non rotating celestial object of mass M has a radius less than 2M, it must be a Schwarzchild black hole with a singularity at r = 0.
But we have derived the Schwarzchild metric in vacuum. Unless the black hole is just the point of infinite density at r = 0, the metric will not hold inside the event horizon all the way till r = 0, because we will have to include the stress energy tensor for the matter that lies inside the event horizon.
While this is still a good physical model outside the event horizon, how can we be certain that there is a singularity at the centre?
 A: I was just 3 minutes ago reading Planck Stars by Carlo Rovelli.  In it he conjectures that a star does not collapse down to a singularity, but rather it collapses down to Planck density: 10^93 g/cm^3 with a size of approximately 10^-12 meters cubed which is still some 24 orders larger than the Planck size.
He goes on that in its own proper time, the Planck star lives an exceedingly short life before it bounces back.  That life is the time required for light to cross the width of a proton, but because of the incredible time dilation would be about 14 billion years of Earth time.  Because this happens to match the current age of the universe, perhaps some of these "bounces" will be detectible to us.
Rovelli's concept helps to solve the problem of information loss.
A: 
how can we be certain that there is a singularity at the centre

You are starting from a Schrwazchild black hole model which is an idealized eternal model and so never changes.  This is a useful basic model of a black hole but it's not how real black holes are - they clearly cannot be eternal as the universe didn't exist at one point previous and indeed the originating stellar object did not either.  We also know black holes expand as they absorb material so that's another issue.  Hawking also showed that they can emit radiation (albeit not a lot) so an eternal black hole model is clearly not a match to real black holes.
While with an idealized Schwarzchild black hole you can state there is a singularity at the center (because that's just the math and it's an ideal), what is inside the event horizon of a real black hole is probably another thing entirely.
Even the next step up from a Schwarzchild black hole (which is non-rotating) to a rotating black hole (the Kerr metric) produces a requirement for a ring singularity and other features not in the Schwarzchild model.
We also don't have a complete and generally accepted theory of gravity that incorporates quantum theory so how matter and energy behave under such extremes as a near-singularity are currently not understood.  It's not at all clear a "pure" singularity could exist, nor what could be in it's place.
So with that in mind we don't really know what the interior of a real black hole is like.  We probably won't ever know as by definition inside a black hole is beyond the side of the event horizon we want to be on.  Unless there is a way around the no hair theorem we might never know.
