It will all depend on who you rent through. If you rent directly from an owner, they may have more lenient standards than an owner that is having their property managed through a property manager or listed through a broker who will screen prospective tenants for them.
If you have credit issues, it is better to bring that up in the first conversation so that both sides can determine if it is going to be a problem. Because of fair housing laws, property management companies and apartment complexes have to apply the same standard to everyone. This means there is very little flexibility for individual circumstances. A potential tenant will either meet the criteria or they won't. If they don't, than there usually isn't anything the agent can do to help them.