There are really a lot of models that would fit your bill. If you can wrench and you really want to save money, then you probably want to go compact Japanese in the $1,000-2,000 range. Check the repair data on this site for whatever model you can find cheaply and make sure that rates of needing repairs are low and that common repairs are ones that don't frighten you. You might have OK luck with Craigslist too, especially if you buy from a real person who drove and took care of the car. Granted, the Japanese compacts don't go quite as cheaply as the big domestics, but there's a reason for that (lower cost of ownership once you've bought them).
I have personal experience with Proteges, Corolla/Prizms and Honda Fits. They're all great cars; especially the Fits. Maybe you can turn up a high milage 07 or 08 for a few grand. If you look, you'll see a lot of the repairs they require are new coil packs, which you would be able to replace with OEM for $200 and an hour of your time; so not a huge deal. The Proteges have a great drivetrain and nice handling, but rust like crazy.
In this day and age, I'm not sure there is such a thing as a death trap. Actually, on Jalopnik there was recently an article on a frontal offset crash test of Nissan's base model in Mexico (a 92 Sentra; they just kept making them down there) going head to head with their base model here (Versa). The Mexican car got 0 crash safety stars; a true death trap. The Versa did OK.