There are frameworks in physics, dimensional and energetic frameworks.
There is the classical framework which has classical theories of mechanics and electrodynamics etc, where the dimensions are compatible with the meters/seconds/kilograms measures.
There is the quantum framework which has quantum mechanics, quantum electrodynamics and quantum field theory. Its framework is in dimensions less than a nanometer and energies compatible with h_bar~1*10^-34 J second .
There is the special relativity framework adjacent to the quantum mechanical framework
There is the general relativity framework for dimensions in light years and masses of the order of stars.
Each framework is mathematically modeled with differential equations and solutions to these differential equations. The potentials and the solutions because of the nature of mathematical equations can have singularities.
For example the 1/r potential of classical electricity and magnetism leads us that at r=0 there exists a singularity for the classical problem. No hydrogen atom. But to go to r=0 the dimensions go to less than a nanometer, and the appropriate framework is the quantum mechanical framework, and lo and behold there are no singularities and we have the atom in a stable quantum state.
When a general relativity solution gives a singularity, again the dimensions go to the framework of quantum mechanics. It has no meaning to worry in terms of general relativity. Unfortunately the gravitational field has not been consistently quantized, there exist effective field theories but still one cannot know what happens at the corresponding r=0 . One can handwave with the heisenberg uncertainty principle, but until there exists a consistent theory of quantized gravity the statement is "we do not know what happens at the singularity", we expect that soon the theory will be extended so that a quantized model of gravity will apply and there will no longer be a singularity in the classical sense.
In the Big Bang model effective quantization has been introduced immediately after the general relativity solution singularity, with the inflaton field , and the cosmic microwave data fit the model. Patience, the mills of physics may grind slowly but they grind exceedingly fine.