How can the universe become infinite in spatial extent if it started as a singularity, wouldn't it take infinite time to expand into an infinite universe?
If the Universe is spatially infinite, it always had to be spatially infinite, even though the distances were shortened by an arbitrary factor right after the Big Bang.
In the case of a spatially infinite Universe, one has to be careful that the singularity doesn't necessarily mean a single point in space. It is a place - the whole Universe - where quantities such as the density of matter diverge.
In general relativity, people use the so-called Penrose (causal) diagrams of spacetime in which the light rays always propagate along diagonal lines tilted by 45 degrees. If you draw the Penrose diagram for an old-fashioned Big Bang cosmology, the Big Bang itself is a horizontal line - suggesting that the Big Bang was a "whole space worth of points" and not just a point. This is true whether or not the space is spatially infinite.
At the popular level - and slightly beyond - these issues are nicely explained in Brian Greene's new book, The Hidden Reality.