The following banks and credit unions offer excellent features and benefits. While we recommend them, they are not our top picks for the simple reason that they are not guaranteed banks. They have however helped a lot of people, and may be able to help you too.
Equitable Savings and Loan Association
They run a True Alert check that only verifies your SS# and address. They do not run a credit report unless you apply for a debit card.
Many people have been able to get accounts with TCF. They use ChexSystems but are extremely lenient, definitely worth a chance.