There are zero contradictions between quantum mechanics and special relativity; quantum field theory is the framework that unifies them.
General relativity also works perfectly well as a low-energy effective quantum field theory. For questions like the low-energy scattering of photons and gravitons, for instance, the Standard Model coupled to general relativity is a perfectly good theory. It only breaks down when you ask questions involving invariants of order the Planck scale, where it fails to be predictive; this is the problem of "nonrenormalizability."
Nonrenormalizability itself is no big deal; the Fermi theory of weak interactions was nonrenormalizable, but now we know how to complete it into a quantum theory involving W and Z bosons that is consistent at higher energies. So nonrenormalizability doesn't necessarily point to a contradiction in the theory; it merely means the theory is incomplete.
Gravity is more subtle, though: the real problem is not so much nonrenormalizability as the high-energy behavior inconsistency with local quantum field theory. In quantum mechanics, if you want to probe physics at short distances, you can scatter particles at high energies. (You can think of this as being due to Heisenberg's uncertainty principle, if you like, or just about properties of Fourier transforms where making localized wave packets requires the use of high frequencies.) By doing ever-higher-energy scattering experiments, you learn about physics at ever-shorter-length scales. (This is why we build the LHC to study physics at the attometer length scale.)
With gravity, this high-energy/short-distance correspondence breaks down. If you could collide two particles with center-of-mass energy much larger than the Planck scale, then when they collide their wave packets would contain more than the Planck energy localized in a Planck-length-sized region. This creates a black hole. If you scatter them at even higher energy, you would make an even bigger black hole, because the Schwarzschild radius grows with mass. So the harder you try to study shorter distances, the worse off you are: you make black holes that are bigger and bigger and swallow up ever-larger distances. No matter what completes general relativity to solve the renormalizability problem, the physics of large black holes will be dominated by the Einstein action, so we can make this statement even without knowing the full details of quantum gravity.
This tells us that quantum gravity, at very high energies, is not a quantum field theory in the traditional sense. It's a stranger theory, which probably involves a subtle sort of nonlocality that is relevant for situations like black hole horizons.
None of this is really a contradiction between general relativity and quantum mechanics. For instance, string theory is a quantum mechanical theory that includes general relativity as a low-energy limit. What this means is that quantum field theory, the framework we use to understand all non-gravitational forces, is not sufficient for understanding gravity. Black holes lead to subtle issues that are still not fully understood.