On a personal level, I would contact your city and see if they have an apartment inspector service. My city does, and you can pay a small fee (I think it's something like $50-$100, depending on how big the apartment is here) to have a person who's properly trained in building and renter's codes to make sure that everything is up to code for you. In most cities, this includes making sure that there's no mold, dry rot, etc, and that everything is in good repair. Also, it gives you a document for your protection if you, say, choose to move into an apartment with a dent in the wall: Your landlord can't try to say that you caused the dent and have to pay.
When I move to a new place, here are questions that I'm sure to ask:
1) Is smoking allowed here?
2) Are pets allowed?
3) Is there carpet?
4) Have you had any leaks or water pipe problems lately? (either can tip you off to the risk of mould)
5) When was the roof last redone? (anything longer than 15 years, and you're risking leaks unless the property manager is a stickler for maintenance)
6) How old is this building? (older = more dust, possibly more mold, and the possibility of asbestos since where I live, people aren't required to tear it out if it's already there; they're just required to not build with it and close off areas that have exposed, damaged asbestos... this may be different where you live).
7) Why did the previous resident leave?
Also keep an eye out when you're touring for stuff like superficial paint jobs, loose molding, old windows in a freshly-redone apartment, patchy paint, and anything else that looks like the apartment manager was trying to make the place look good on the cheap. That suggests they may cut corners regularly.
Thanks for all of the thoughts! Sorry I didn't write back sooner...I've been packing (and had to go and ask for inhalers). I guess my lungs don't like dust even if I'm not supposed to be allergic to it.
Angel