Yes, payday loans are dischargeable in chapter 7 and chapter 13 bankruptcy. A person can easily get sucked into a vicious vortex of debt, taking out one payday loan to pay off a prior one. The important thing to do, is to ensure that you have good mailing addresses, and that while your case is being prepared for filing, that you cancel any permission you may have given them to take money out of your bank account or to do a voluntary wage withholding.
There is always a slight possibility that substantial borrowing activity shortly prior to filing could lead to a lender objecting to the dischargeability of that particular debt. “Substantial” and “shortly” are a matter of perspective and depend on individual facts. I have never seen a payday lender make such an objection though.