I hope some experienced folks respond to your post. Since no one has yet, I'll tell you what we did in your situation, although we aren't experienced doing adoptions. When I spoke to the personal references, I basically tried to get a sense of the potential adopter's world--their schedule, their social group, their job, what their life is like. Part of that was looking for confirmation of what the adopter had said in their application. (Of course you still need to directly fact-check things like the landlord, the vet, etc.) Ask open-ended rather than yes-or-no questions; try to draw the person out and get them talking. If the reference is a vet or an employer, you can be pretty direct in asking about how responsible the person is, etc.--would they let this person care for their pet? If it's a friend, you can ask them to describe what kind of owner their friend would be, what kind of life a dog would have with them. You also end up getting a sense of whether the reference-person is trustworthy, which says something about the judgment of the adopter. A lot of the information I gleaned was not only literal facts, but also vibe & gut feeling about the person.
One note: it can be interpersonally delicate. I found that people could be a bit defensive---feeling like their friend was being overly checked-up-on and that the questions were overkill. It's good to try to defuse that a bit up top by having a sunny demeanor (not seeming like an interrogator), and by giving a short explanation at the beginning about why you're talking to the reference. ("We try really hard to make a great match between adopters and dogs." "Your friend seems like a great applicant, and we wanted to learn more about her." The fact is, you ARE interrogating them for the maximum amount of accurate information, but it's good to disguise that with charm and positivity as much as possible.
Good luck and I hope you get some more responses.