There are sevaral reasons why singularities may not exist. Einstein's field equations are continuous functions of space-time, and so admit the possibility of a singularity. Interestingly enough, proving that a singularity exists in a black hole (or at the start of the Big Bang) requires assumptions about the energy contained within a black hole (or assumed at the instant of creation). These are called dominant energy (also called strong energy) assumptions. These assumptions are that 'normal' energy will dominate over any possible negative pressure created by scalar fields. Both Hawking and Penrose proved that under the assumed energy conditions that all geodesics within the black hole will be incomplete, meaning that they will terminate at point in space-time. This gives the singularity a little more meaning - as a discontinuity - rather than infinite curvature as would be inferred from the infinite value ot the so called curvature invariant at R=0 derived from General Relativity. Many now think (Hawking included, not sure what Penrose thinks these days), that the energy assumption is invalid. Note that viewing the singularity as a discontinuity is somewhat consistent with the idea that space-time is not continuous, but not necessarily quantized in the vein of quantum field theory.
So, not only is the dominant energy assumption in the above not necessarily true, there is also a modification to general relativity that can do away with a singularity. It is called the Einstein-Cartan theory and it adds quantum spins as a source of gravity in general relativity. This is pretty complicated but the original theory is non-torsional. The components of the metric tensor (solutions of the field equations) are symmetric and 'twists' in space-time are not allowed. The torsion generated by the quantum spins introduce torsion in the space-time metric that actually generates a repulsive gravity effect. So, the idea goes that within a black hole, as things get compressed to an extraordinarily small spaces, densely packed quantum spins will exert enough negative gravity to stop the formation of a singularity.
The Einstein-Cartan theory isn't a quantum gravity theory (the field is not quantized) and is much more difficult to solve than the original theory because of the lack of symmetry. Testability is questionable since there is no difference between Einstein-Cartan general relativity and Einstein's general relativity except at the actual source of the quantum spins (the vacuum solutions are identical).
In any case, the above are reasonable theories for eliminating the very non-reasonable singularities of vanilla general relativity under dominant energy conditions. Neither really require a quantized space-time to do away with a singularity. Quite frankly, I don't know that it is provable that a quantized space-time does away with the singularity. Not sure there is much conceptual difference between stuff going into a Penrose discontinuity or stuff going into a discontinuity between two quantized points in space-time.
Edit: quick update to a rather long and probably boring answer. Most gravity quantifications assume the existence of gravitons as quanta of gravity. In the weak field approximations where you can in fact model gravity with gravitons, the gravitons are assumed to exist in a flat continuous space-time. So, a true quantum gravity may not need a quantized space-time.