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.