I would only consider single speed for a 5 day/week 20 mile round trip if you are (1) already in outstanding shape and (2) have no hills on your commute.

There are probably a lot of good used bikes out there, although it might be hard to evaluate what is "good" if you don't know bikes already. If you find a good deal on a used bike, you might want to put any leftover money from your budget into having a bike shop do a complete tune-up. Hopefully this would include rebuilding bearings like hubs, headset, and bottom bracket that might cost you more money down the road if they fail. Bearings and grease for a hub cost a couple bucks, but even a cheap new wheel will cost at least $50.