Some landlords do not check credit. They meet the possible tenants and decide in person if there is a match to their property and their ownership and the tenant. For those landlords, dress nice and leave the pit bull at home. For those landlords who do check credit, have good and reasonable explanations and proof that the problems are being/have been corrected.
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It depends on who is doing the screening and the requirements of a landlord. Some landlords want 700 credit scores and others are willing to do lower. The process for the screening is again dependent on who is doing the screening. Typically though it's 3-4 days. It shouldn't take longer than that unless it's hard to get a hold of people.