We found our dog through petfinder.com. The rescue orgs that list through them seem to do the best they can to provide the dog's history, and whether they have specific needs regarding their health and behavior. For example, they will indicate whether a dog would get along with other animals in the house or not, or whether or not a dog would be good with children. I think they try to be thorough in the interest of finding a good fit.

Breed rescue might also be something to explore. I don't know much about it, but if you are interested in a specific breed, there are organizations that specialize that way as well.