Singularity Avoidance in Canonical Quantum Gravity
The question of whether quantum gravity has something to say about spacetime curvature singularities has a long history, from the early investigations in the 1960's, to more quantitative work on symmetry reduced models in the 1970's using the Wheeler-DeWitt quantization. Some examples of this include work by Misner , and by Blythe and Isham  on Friedmann-Robertson-Walker models with a scalar field.
After a lull of several years, the canonical quantum gravity programme was revitalised by the Ashtekar's triad-connection phase space variables for general relativity. The variables naturally represented as operators in this so-called loop quantum gravity approach is the densitized inverse triad, and the holonomy of its conjugate connection variable.
The first significant development concerning singularity avoidance in the loop quantum gravity (LQG) programme occurred inadvertantly, in an attempt to define a regularized Hamiltonian constraint operator. This was the construction of a triad operator by Thiemann . It was realized a few years later by Bojowald  that the algebraic relation used to define this operator in the full theory could also be used in quantum cosmology to define an ``inverse scale factor'' operator. The ``source'' of singularity avoidance in the symmetry reduced models studied so far in ``loop quantum cosmology'' (LQC) is that the inverse scale factor operator is bounded, at least in the isotropic models.
The basic mechanism used in defining the inverse scale factor operator in LQC may be illustrated in a mechanical system. Consider a 2-dimensional phase space with planar topology, and consider a quantization on a spatial equi-spaced lattice with spacing . There is a quantization and a basis such that the translation is realized as a shift operator, and the position is diagonal. An inverse operator can now be defined by starting with the Poisson bracket identity
The Poisson bracket ``trick'' used for the singularity avoidance result is applicable in other models for quantum gravity outside the LQG context; it does not depend on the use of the connection-traid variables which are at the basis of LQG, but rather on the choice of representation used for quantization. This observation was exploited by Husain and Winkler to revisit the quantization of models systems formulated using the ADM variables. This has led to singularity avoidance results for FRW models which are qualitatively similar to those found in LQC, and also for the gravitational collapse problem in spherical symmetry .
There are two aspects of the results of singularity avoidance. The first is kinematical in the sense that the operators corresponding to the inverse triad in the model systems is bounded. The second is dynamical in that Hamiltonian evolution is well defined and unitary beyond the point of the singularity - that is evolution does not terminate there. This feature of evolution in LQG based models arises due to its innate lattice structure: the Hamiltonian constraint acts discretely so that the Hamiltonian constraint condition is a difference equation rather than a differential equation as in Wheeler-DeWitt quantization. For regions away from the classical singularity, the difference equation behaves merely like a discretisation of the corresponding Wheeler-DeWitt equation, but for regions close to the singularity this is not the case, due partly to the incorporation of the inverse scale factor operator in the difference equation.
This second aspect was demonstrated explicitly by Ashtekar, Singh and Pawloski in an FRW model coupled to a massless scalar field . By considering evolution using the scalar field as a time variable, they showed using a numerical computation that a wave packet loses coherence as it evolves toward the classically singular region, and after a bounce begins to regain coherence as it evolves away from this region.
Beyond model systems, there has been a detailed investigation by Brunnemann and Thiemann of triad operators in full LQG . One central result here is that such operators are not bounded above in the full theory in a strict sense - there are certain classes of states in the kinematical Hilbert space of LQG that lead to the result. While the inverse scale factor in anisotropic LQC models is also unbounded, its eigenvalues on zero volume eigenstates is bounded. In full LQG, however, even on zero volume eigenstates the operator corresponding to the inverse scale factor is unbounded.
This work brings to the fore in the LQG setting the old question of the relevance of mini-superspace quantization for understanding quantum gravity. Specifically to the issue of singularity avoidance, it raises the question of what features of the full theory are ultimately responsible for this, and how it manifests itself in symmetry reduced models. For instance, the second conclusion drawn from  is that the expectation value of the inverse scale factor operator remains bounded in the sense of expectation values with respect to a one parameter family of coherent states whose peak in phase space follows the classically singular trajectory. This result of full LQG implies a completely different sense of singularity avoidance than the one obtained in LQC, but at least it does not contradict the LQC result.
Further understanding of the issues raised by this work entail going beyond the simplest models, to at least those that have some inhomogeneity. This is being studied by several people both in the context of cosmology and gravitational collapse.