Singularity - exactly what is it? In quantum mechanics and gravity theories one always meets the word "singularity" in connection with talks about black holes and the big bang, etc. Now in mathematics a singularity is well defined – mostly as a pole of $n$-th order that makes the function value approach infinity near the singular point...
But I cannot imagine that a singularity can actually exist in the real world. It is simply "un-physical" and any equations having physical meaningful singularities must be wrong or the singularity somehow excluded just like saying $v/c$ is always less than 1.
Hence, a black hole can not in it self "be" a singularity but might behave as such near that point, which, however, nothing ever reaches. Thus it can not be observed and is physically non-existent as a point in time and space but only an abstraction – similar to what I would be tempted to say about the photon.
 A: The formal definition of a singularity in GR is not simple, and there are various definitions which are not completely the same.
Intuitively a singularity is a place where some physical quantity becomes ill-defined: for GR that quantity is almost always curvature, since that's the physical quantity GR is interested in.
However a definition that actually ends up being used a lot is that of 'geodesic incompleteness', which means that there are geodesics (more generally: suitably smooth parameterized curves) which can not be arbitrarily extended.  This definition abstracts the idea that, for a black hole solution, there are timelike geodesics which can only be extended for finite proper time.  However this definition doesn't really talk about why the geodesics are incomplete, just that they are.
Another definition involves the notion that there are regions which need to be cut out from the manifold somehow, and this is related to the previous one since there will be geodesics that intersect these regions which have been cut out of the manifold.  But again, it doesn't say why they were cut out.
There are probably other definitions.  Note that, using the last definition, you can always construct a singularity just by cutting bits out of the manifold (see below).
The Hawking-Penrose singularity theorems are phrased in terms of geodesic incompleteness.  I can't now remember if they are stronger than that: are the singularities necessarily curvature singularities?  I suspect they aren't, but I also suspect that's just because proving that would be too hard, and in practice they are.
I think it's reasonably clear that for the second and third senses, that if physical quantities are well-behaved at the singularity then you could always somehow extend the manifold past it, constructing a larger manifold which did not have the singularity: this would certainly deal with the constructed singularity case I mentioned above.  However I might be wrong that you can always do that (it's hard to see what could go wrong doing this though: possibly some global topological problem).
I think if you use the informal 'divergent physical quantity' notion, then yes, it's clear that singularities should not exist in a complete theory.
However it's important to understand that the singularities predicted by GR are fairly toxic: there's no escape in the sense that it might somehow take an arbitrarily long (proper) time to actually reach the singularity so they don't really matter: on the contrary there are timelike geodesics which intersect the singularity (and are therefore incomplete) in finite and, in practice, small values of proper time.
I'm not aware of singularities in QM.
A: The singularity exists only within the theory of General Relativity alone. But it is well known that at this scale and energy, Quantum Mechanics must play as well, and uncertainity principle is not compatible with singularities. The problem is that nobody knows yet how to couple these 2 theories, it's one of the biggest challenges of nowdays physics ( And of course we probably won't experiment or go verify on place as well. :-) ). 
So in the real world there is very likely no singularity, but nobody can tells what is there instead.
A: 
I cannot imagine that a singularity can actually exist in the real world.

And indeed, it does not because singularity is not a real-world concept:  It's a math concept, and when you hear about singularities in physics, you are hearing about singularities in the equations of some mathematical model of reality.  A singularity in a mathematical function simply is an isolated point or a connected set of points in the domain of the function for which the value of the function is undefined.
Singularity is not only a problem in General Relativity.  E.g., you can Google for "mechanical singularity" for some other examples.
I don't know about General Relativity--that math is far beyond my ken--but in general, when a singularity exists in some model of reality, it usually means that the model is simpler than the reality, and its usefulness "breaks down" in the neighborhood of the singularity.

There's a curious feature of General Relativity though:  GR predicts that an event horizon must exist between any singularity and an observer.  So, it's theoretically impossible to discern the difference (if any) between the reality and the theory.
