First, you'll need the street address of the building in question. (You don't need the zip code.) Go to the Department of Housing Preservation & Development home page. In the right column, enter the address of the building. That should lead you to a page with a lavender column on the left.
Click All Open Violations. There you'll see names of the building owner and management. Beneath that info, in bold, is the number of all open violations, along with a breakdown of class "A" (non-hazardous), "B" (hazardous), and "C" (immediately hazardous) violations. Scroll down and you'll see descriptions of the complaints. For example: "nuisance consisting of vermin mice in the entire apartment located at apt e9" or "remove the encumbrance obstructing egress from fire escapes a/c protruding from window at 3rd story balcony" (i.e. an air conditioner is blocking the fire escape).
Most buildings have open complaints, especially if they're large, so the mere presence of mice shouldn't be a deal breaker. As a rule of thumb, the way to rank a given building is to take the number of open violations and divide it by the number of apartments ("A Units"), which you'll find listed in a horizontal graph at the top of the page).
More than 5 violations per unit: slumlord; avoid
Fewer than 3 violations per unit: average/good
1 violation or less per unit: excellent
As for residents of other cities, check with your equivalent of the NYC housing department to see if their records are online. If they are, please let us know and we'll post about other city's records in the future.
Link: NYC Department of Housing Preservation & Development
(Photo: Stuck in Customs)