diddly-squat

Imagine the universe as an infinite ruler. The ruler was infinite in extent when it came into existence (at t =0) and remains infinite in extent for all very large values of t. Now, as t increases the ruler 'expands'; this doesn't mean that the extent of the ruler changes (remember it's infinite), what it means is that 'distance' between the notches on the ruler increases. This is what is meant by space-time expanding. It's also important to remember that locally gravity dominates, it's only on large scales (inter-galactic space) that the universe expands.

A singularity is a mathematical artefact and thus has no real 'physical' form. It is simply a place where our models cease to work. A good example is that of an even horizon (which is by definition a 'singularity').