Just to add a couple things to the excellent info VelocideX has provided...
First some of your language isn't quite right, though the basic idea is. 'Theoretical physics' is just physics that isn't related to designing / performing experiments. The two fundamental physical theories we have are 1) General Relativity and 2) the Standard Model of quantum field theory. GR explains gravitation and so the behavior of the universe at very large scales. The Standard Model explains all the other forces and so the behavior of the universe at small scales. (Vanilla quantum mechanics is an approximation to quantum field theory.)
Singularities, also, aren't mathematical constants. Generally speaking, a singularity is a point where important quantities become undefined, typically going off to infinity as it is approached. eg, the function f(x)=1/x has a singularity at x=0 . In GR, a singularity usually means a point where the density of matter goes to infinity, and time for an observer simply ends (or begins.) This happens at the beginning of the universe -- there's no meaningful way to say what's going on at t=0 or 'before.' In a sense, only positive times exist. That's very odd, but there's no immediate reason to think that quantum mechanics couldn't be valid at all positive times, thus not making the situation any weirder.
However, as VelocideX noted, there *are* fundamental incompatibilities between the two theories. To begin with, they're not even written in the same language. GR treats all mass/energy alike, as a continuous distribution on a curved spacetime. QFT uses entirely different concepts, which are more complicated and much harder to explain, and is built on a flat space. Clearly, if you want to know what happens when both gravity and other forces are important, the two must be able to be merged in some way. The most obvious way to do this -- reformulating GR in QFT terms, just like the other forces -- turns out not to work.
The main current ideas are 1) to try and rewrite QFT so it's not built it off a fixed spacetime -- this is loop quantum gravity, or quantum geometry -- or 2) assume QFT is just an large-distance approximation to another, more fundamental, theory which includes gravity treated just the same as the other forces... this is string theory and its variants.
There's a good short history of all this at
http://www.arxiv.org/pdf/gr-qc/0006061 , though it likely won't make much sense without a lot of knowledge about quantum gravity to begin with...
GR is near impossible to test, because it only supersedes newtonian gravity in regions of high mass (or high energy)...
Luckily, we can see plenty of astronomical objects that provide plenty of mass.
So there are a lot of experimental tests that have been done, actually... see
http://pup.princeton.edu/sample_chapters/ciufolini/chapter3.pdf and
http://www.arxiv.org/pdf/gr-qc/9811036 for (semi-) recent reviews.
True - TP and QM *are* on solid ground. But your argument about the singularity does not come from theoretical physics; it comes from an extrapolation of general relativistic results backwards in time.
You can also prove (in GR) that singularities must exist at the center of black holes, BTW.
