Housing lease

Look, if you have a bank account, you'll likely be paying your rent with checks. At least, I normally don't carry around 700-1000 bucks in my pocket around the beginning of the month.

Whether you like them having your bank account info or not, they will probably end up being able to read it off of your rent check the moment you pay your deposit or first months rent.

I dunno what to say about the maiden name bit. Just say you don't know it.

Jam it back in, in the dark.

Oh, and don't ever give a retailer your credit card number either. They might steal it and order stuff....

Seriously, people are so paranoid over identity theft. If it's going to happen, there probably isn't shit you can do to stop it. Just periodically check your statements and get your free annual credit checks and you can at least repair damages.
Most, if not all lenders and banks have a Zero liability policy. And somebody who you're actually buying something from (or leasing from) is the least likely place to steal your monies.

Mother's maiden name, if I'm not mistaken, is a common field on a background check application.

There's nowhere I can't reach.