Since the aim of this project is cheapness, use a bike you have laying around. Many bikes will work for this, as long as it has one crucial component, a horizontal ( or horizontal-ish) dropout. While it is possible to convert a bike with any dropout to single speed, it takes either a great deal of time or money, neither of which are appropriate for testing the waters of single-speed bikes.
Other things to look for:
If the bike has horizontal dropouts, it will work, but if you have the choice, get a frame without brazed on cable stays. They will only affect how your bike looks, but appearance does count sometimes, and single speeds look cool, so why mess with it.