There are homeowners out there who does not check rental history and you may still be able to rent. Even if they check and it's settled they have a possibilty of renting to you. I would suggest trying to find a reputable real estate rental company and be careful on Craigslist.
If you are trying to rent an apartment, again some are ok with it and some may not. You can always ask questions before putting in an application and paying the application fee.