Why singularity in a black hole, and not just "very dense"? Why does there have to be a singularity in a black hole, and not just a very dense lump of matter of finite size? If there's any such thing as granularity of space, couldn't the "singularity" be just the smallest possible size?
 A: Because otherwise general relativity would contradict itself. The event horizon of a black hole is where not even light can escape. Below the horizon all photons must fall. In relativity theory all observers measure the speed of light the same, c; that's a postulate of the theory. Then all physical things (including observers) at and below the horizon must fall and keep falling, lest they measure the speed of light emitted upward to be something other than c. If you could stand on a very dense lump of matter of finite size at the center of a black hole, and pointed a flashlight upward, the photons would somehow have to fall to the ground (without moving upward at all) and you wouldn't measure the speed of light to be c in the upward direction. The theory would be broken. The singularity is the "can't fall further" point and the theory becomes inapplicable there.
A: It's important to understand the context in which statements like "there must be a singularity in a black hole" are made.  This context is provided by the model used to derive the results.  In this case, it was classical (meaning "non quantum") general relativity theory that was used to predict the existence of singularities in spacetime.  Hawking and Penrose proved that, under certain reasonable assumptions, there would be curves in spacetime that represented the paths of bodies freely falling under gravity that just "came to an end".  For these curves, spacetime behaved like it had a boundary or an "edge".  This was the singularity the theory predicted.  The results were proved rigorously mathematically, using certain properties of differential equations and topology.
Now in this framework, spacetime is assumed to be smooth - it's a manifold - it doesn't have any granularity or minimum length.  As soon as you start to include the possibilities of granular spacetime, you've moved outside the framework for which the original Hawking Penrose theorems apply, and you have to come up with new proofs for or against the existence of singularities.
A: See Carter 1968 for why rotating black holes that have incoming disturbances may not have a singularity at all. 
A stationary non - rotating hole will have a singularity. But no one thinks that these exist in nature. But with rotation that singularity 'shrinks' to a ring. The set of paths that hit the singularity is shrunk to a mathematical 2D plane from 'all directions' with the Swarzschild Soln. Then with incoming 'noise' it may be that there are no paths - geodesics - that lead to a singularity. 
http://luth.obspm.fr/~luthier/carter/trav/Carter68.pdf
All exact solutions of General Relativity are done with asymptotically flat space, which does not exist in the real world. So while the theory of GR admits singularities, in a real classical GR world they likely don't exist.
Carter actually always talks about a singularity, but one with no paths to it. No ouchy at the end of a path. With no paths to a singularity - is it really there? I would think not, and as Carter points out, others do too. (Lifshitz and Khalatnikov).
A: The chosen answer is quite good. This is a general answer for an unsophisticated audience whose naive questions are sent here as a duplicate, as in this case.
Classical physics developed when calculus and differential equations entered the field and made possible the mathematical modeling of observations and data; before the times of Newton the models had not advanced further than using algebra and euclidean geometry. 
The mathematical formulas appearing in classical physics are  full of singularities. Take the 1/r potentials in electricity and gravity. The approach to r=0 predicts larger and larger fields, up to infinity. This is not a problem because classically any object has a volume, no matter how small, and it was understood that the infinities were theoretical extrapolations, for the classically non existent states of point particles. Any particles were presumed to have a mass that could not be compressed to a point, so these singularities were not a problem. When experiments started getting data below the nanometer level, Quantum Mechanics had to be invented in order to explain the data, and quantum mechanics comes with the Heisenberg uncertainty principle,HUP, which turns all singularities into a fuzzy region . The electron does not fall on the proton but is constrained to orbit around it. The same with the electron on the positron. Free  electrons are posited to be zero point particles with mass, but there is no infinity in the field due to the HUP, active in any interaction that would define r.
General relativity is also a mathematical model for very large scales and energies and mass, for the gravitational observations. As the chosen answer shows a mathematical singularity is extrapolated in the mathematical description of classical black holes.
The same is true for the original mathematical model of the big bang cosmological model, where a different type of singularity was postulated using General Relativity, where all the presently seen energy of the universe appeared. Astrophysical observations  forced the conclusion that at the very beginning quantum mechanics has to be used, so the beginning of the universe is a fuzzy region and the mathematical singularity non existent.  ( we are still waiting for a definitive quantization of gravity though).
So the title:

Why singularity in a black hole, and not just “very dense”?

can be answered by: It is not a singularity but the concept of "dense" is quantum mechanical, "dense probability distributions for the energy content" generating a quantum mechanical  mathematical fuzziness around the classical singularity point.
A: For any experiments a spherical Black Hole behaves the same way as if its mass was uniformly distributed over its surface or uniformly distributed over its volume or concentrated in its center. These variants are indistinguishable.
It is impossible to find exact distribution of mass inside a black hole because it has no internal structure, due to holographic principle (if it had, it would be possible to transfer information out of black hole via gravitation waves).
A: My understanding is that the Uncertainty Principle forbids point masses, which would have 0 uncertainty in position, and thus total uncertainty in momentum. The well-known result is that no particle can be confined in a region smaller than its wavelength. Three or more solar masses in the space of one particle is indeed a very high density, but not infinite.
It is the same as the reason why electrons, with a much longer wavelength than protons and neutrons, cannot fall into the nucleus of an atom. They are already as close as they can get.
A: My two cents; singularities do not need to form. They are for all intents and purpose, a rough model that neglects quantum physics. 
If you have the right kind of argument, you can have models that avoid singularities. The event horizon, is not a true singularity but is in fact a coordinate phenomenon (ie. space becomes timelike and time spacelike). There are today, models which attempt to explain the collapse of a star in such a way, that singularities do not form. 
A: Actually there are no singularities inside the black hole. It is just a mathematical special point on the coordinate system, which does not correspond to any real-world singularities. 
The event horizon of a black hole is just an impenetrable barrier on which the time is frosen so nothing can pass it. In another model the black hole as a whole behaves as a viscous liquid with quite limited density (the density decreases as the BH's mass rises).
