What paperwork do I need to convert a 30-day visa exemption to a non-immigrant visa in Bangkok?

Okay, what you need to do is FIRST make sure you have at least 15 day or MORE left on your current 30 day visa exempt entry, 60 day tourist visa entry or a 30 day extension from one of those stamps.

If you do not have 15 days or MORE left on your current stamp you cannot convert to an in country Non-Immigrant Type visa

IF you do have 15 days on your stamp GO out to the bangkok immigration office out at the government complex on Chaengwattana Road, GO to Section C-1 <- the section that does in country conversions of the stamps I mentioned to 90 day Non-Immigrant Type visas and get the hand out for the reason you're applying for the 90 day Non-Immigrant visa. (They have different hand outs for marriage to a thai, being over 50 (retirement), working here, teaching here, education, etc) Get the hand out for the reason you want the visa, and start compiling the requirements as far as the documentation.

Good Luck, one last time, GO to the main thai immigration office at the Chaengwattana Government Complex Building B, first floor south entrance and go to section C-1.

Perhaps you should be a little more specific about yourself and which visa you're after?

Let us look in to the crystal ball