I'm only assuming that you're talking about renting since a non-financial background check wouldn't matter to a mortgage lender. What I do know is that Madison is very tenant friendly even despite some changes made to landlord-tenant laws by the state.
There are multiple resources in print and online that will show you what's available for rent. "Good neighborhood" means different things to different people. Typically, I like to ask a beat cop about certain neighborhoods since they are nowhere near bound to restrain their opinions about areas as Realtors are. Good luck.