Question: Has the problem of quantum gravity and the reality of spacetime singularities been solved?
About the article "General relativity", section "Current status".

Candidate passages:
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests.
However, there are strong indications the theory is incomplete.
The problem of quantum gravity and the question of the reality of spacetime singularities remain open.
Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.
Even taken as is, general relativity is rich with possibilities for further exploration.
Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations, and increasingly powerful computer simulations (such as those describing merging black holes) are run.
The race for the first direct detection of gravitational waves continues, in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.
More than ninety years after its publication, general relativity remains a highly active area of research.
Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result.
The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail.
Can quantum mechanics and general relativity be realized as a fully consistent theory (perhaps as a quantum field theory)?
Is spacetime fundamentally continuous or discrete?
Would a consistent theory involve a force mediated by a hypothetical graviton, or be a product of a discrete structure of spacetime itself (as in loop quantum gravity)?
Are there deviations from the predictions of general relativity at very small or very large scales or in other extreme circumstances that flow from a quantum gravity theory?The theory of LQG is one possible solution to the problem of quantum gravity, as is string theory.
There are substantial differences however.
For example, string theory also addresses unification, the understanding of all known forces and particles as manifestations of a single entity, by postulating extra dimensions and so-far unobserved additional particles and symmetries.
Contrary to this, LQG is based only on quantum theory and general relativity and its scope is limited to understanding the quantum aspects of the gravitational interaction.
On the other hand, the consequences of LQG are radical, because they fundamentally change the nature of space and time and provide a tentative but detailed physical and mathematical picture of quantum spacetime.
Presently, no semiclassical limit recovering general relativity has been shown to exist.
This means it remains unproven that LQG's description of spacetime at the Planck scale has the right continuum limit (described by general relativity with possible quantum corrections).
Specifically, the dynamics of the theory is encoded in the Hamiltonian constraint, but there is no candidate Hamiltonian.
Other technical problems include finding off-shell closure of the constraint algebra and physical inner product vector space, coupling to matter fields of Quantum field theory, fate of the renormalization of the graviton in perturbation theory that lead to ultraviolet divergence beyond 2-loops (see One-loop Feynman diagram in Feynman diagram).
While there has been a recent proposal relating to observation of naked singularities, and doubly special relativity as a part of a program called loop quantum cosmology, there is no experimental observation for which loop quantum gravity makes a prediction not made by the Standard Model or general relativity (a problem that plagues all current theories of quantum gravity).
Because of the above mentioned lack of a semiclassical limit, LQG has not yet even reproduced the predictions made by general relativity.
An alternative criticism is that general relativity may be an effective field theory, and therefore quantization ignores the fundamental degrees of freedom. 
The topology of spacetime is very complex and mathematical, but it can be summed up by saying that spacetime (at least from a general relativistic view) is globally and locally smooth, flat and continuous.
However, when one factors in quantum fluctuations (of the form formula_11), spacetime is not locally smooth.
Such areas as quantum gravity and string theory are trying to resolve this fundamental problem between quantum mechanics and general relativity. 
Lasenby, Doran, and Gull of Cambridge University have proposed a new formulation of gravity, termed gauge theory gravity (GTG), wherein spacetime algebra is used to induce curvature on Minkowski space while admitting a gauge symmetry under "arbitrary smooth remapping of events onto spacetime" (Lasenby, et al.
); a nontrivial proof then leads to the geodesic equation, formula_69 and the covariant derivative formula_70, where ω is the connexion associated with the gravitational potential, and Ω is an external interaction such as an electromagnetic field.
The theory shows some promise for the treatment of black holes, as its form of the Schwarzschild solution does not break down at singularities; most of the results of general relativity have been mathematically reproduced, and the relativistic formulation of classical electrodynamics has been extended to quantum mechanics and the Dirac equation. 
There are other points of tension between quantum mechanics and general relativity.
First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent with general relativity in the neighborhood of singularities (however, no one is certain that classical general relativity applies near singularities in the first place).
Second, it is not clear how to determine the gravitational field of a particle, since under the Heisenberg uncertainty principle of quantum mechanics its location and velocity cannot be known with certainty.
The resolution of these points may come from a better understanding of general relativity.
Third, there is the Problem of Time in quantum gravity.
Time has a different meaning in quantum mechanics and general relativity and hence there are subtle issues to resolve when trying to formulate a theory which combines the two.