I really don't know how to screen people, which worries me. The people who have responded so far - and the ones I'm considering - seem all right. But I have tried to be sensible: I didn't give out the adress or description of the building (just the nearest intersection) and when I schedule a tour with somebody I plan to have them meet me outside a bar nearby, not right at the apartment.

What you said about checking character references seems like a good idea. What's a polite way to ask for a reference?