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.
To provide a simpler answer, one must understand that the observable universe is but a small section of a much larger whole. As light has a finite speed, we can only observe features that have had sufficient time to send us that light. We can only see features that are about as old as the universe itself. Space, however, expanded itself beyond just the small region in which we exist. We cannot see anything in the rest of the universe as light from those areas has not reached us. The other problem is that when we observe something 14 billion light years away, we are only seeing how that feature was 14 billion years ago. So we cannot know how those same features look at the present as nothing is ever observed truly in the present. Even your own reflection in a mirror is only a representation of how you looked in the very recent past as it took a few fractions of a second for the light from your reflection to reach your eyes. INFINITY, however, seems difficult to exist as an actual measure of space. We cannot know if the un-observed universe extends into infinity though this concept itself is the last part of the problem. It is quite meaningless to state that the universe extends into anything as there is nothing for the universe to extend relative to in the first place. When people state that the universe is infinite, they are not so much speaking of size, but of the very shape and nature of space itself. It would be a feature of space that one could travel in a straight line for an infinite amount of time. Actually, there are models that suggest that were one to depart a location and somehow travel in a straight line one would eventually circumnavigate the universe and return to their origin but be a reversed version of themselves - a mirror image - upon their return. Thos has to do with the probability that our familiar 3 dimensions (up-down, left-right, forward-back) are only illusions, and that there are two more spatial dimensions (which are called anna-katta). You move through these extra dimensions without realizing it, but they have a consequence over very large scales, such as a theoretical circumnavigation of the universe. The point is that what we see and experience as "the universe" is only the tip of a much bigger and more complicated iceberg. What we CAN observe is also limited and a consequence of our limited comprehension and our physical inability to observe much of the complexity. It is much like how a fish cannot comprehend the surface of the moon or even suspect that other worlds exist. Their environment and physical limitations, like ours, prevents them from seeing the "big picture".