You are usually refused based on ChexSystems, rather than your credit report. Banks report to ChexSystems when customers have problems making payments, and as a result the customer will have a report made against their name. This means that when they next go to apply for a new account they will likely be turned down.
If you find that you have been reported, there are a series of steps that you should follow: Do not close the account, and make sure you keep your balance in the positive, pay off any outstanding overdraft fees as soon as possible, and make sure any checks you have issued will clear.
If you have been refused you are able to view your report for free, by visiting chexhelp.com. If entries have been made in error, you can write and request that the information be amended. Entries are automatically removed after a specified period of time, usually five years.
If waiting this period of time is not an option, and you are not able to remove a previous bad record from your report then you may wish to apply for a bad credit option. These are issued by a huge number of banks, and it pays to do your research and look at a few different products before committing.
Many of these are advertised online, but it is hard to find any information on them (other than on their own websites) - so it is hard to know how reputable these companies are before you sign up for any of their products.
Another option is to find a local bank and occasionally they will approve you. Additionally you can ask a friend or family member to co-sign with the bank for you. However make sure you intend on keeping your finances in order because you would hate to cause the other person to be listed in ChexSystems because they were trying to help you.
The co-signer is 100% responsible for your account and their own. Another option is to get a prepaid card. This will often give you free direct deposit and the better cards will provide you with free online bill pay. In other words it will give you access to everything you would get with a checking account.