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The new name for the American Institute of Philanthropy is "Charity Watch." They have a short list of national shelter programs with a high rating (B or above). Here is the list.
Note that they do not endorse individual shelters in each city or town, but only umbrella organizations.
Charity Navigator has a list of 66 homeless shelters across the country that have a 4-star rating. Here is the list: http://www.charitynavigator.org/index.cfm?FromRec=1&bay=search.results&cgid=6&cuid=28&sortby=rtg
The BBB doesn't have national lists of homeless shelters to check out. They work in the opposite direction: you enter the name of a charity and they will tell you if they have information on it or not. There is more information here: http://www.bbb.org/
Please let me know if you have questions.
All the best,
Please note that the rating is solely for my service in searching for an answer to your question. It does not have anything to do with what information companies and businesses make available online, since I have no control over that.