I have worked with various real estate ventures and can tell you it depends on the landlord or property management group standards.
If you have no current civil judgments against you and no prior evictions - you should be able to get an apartment with little or no difficulty.
The more upscale - the higher your score will likely have to be.
Be sure to search for an apartment that matches your income. This will be your best shot. If your income is $2000/month - a $1000 a month apartment will be a stretch and your application will more likely be denied.
Apartment managers I have worked with in the past have accepted many people with poor credit if they pay a higher deposit. This may or may not be your case. Having the appropriate cash on hand will always be helpful. Money talks and it's difficult to negotiate a poor credit score if you don't have the money on hand.
When a previous partner and I would rent to people - we'd accept low scores (in the 500's even) if the person could explain. We had one woman who had 5 months of rent on hand for a 6 month lease - she got the apartment without a credit check.
*Find apartments within your budget
*Talk to the leasing agent/landlord before filling out the application
*Be sure to have all of the appropriate documentation
*Save for the security deposit - many landlords require first and last month - regardless of your score - have the money READY (don't want to miss a great deal)
*Be honest on your applications
*Pull your own score beforehand so you can know what they're going to see (go to truecredit.com and try them - get all three bureaus)
*Enjoy searching for your apartment! Use resources like Craigslist to find a place
*If you DO have evictions or unsettled civil suits on your credit try to rent from people who own their own property - they may be more lenient
Bottom line is to work on building your credit. It's your reputation. It affects your entire life.