I put my school on a 0% credit card -- way cheaper than student loans!
I think most schools are charging fees now, though. There used to be a scam where you could massively overpay you tuition bill on a credit card and then the school would cut you a check for the difference. (Lots of people legitimately end up overpaying because grants or loan dispersements can hit your account at weird times)
The interest on that expense would completely wipe out the benefit of frequent flier miles or cash-back and would stick you with a drastically higher total payment (credit card rate =~ 19%, student loan rate =~ 5%). If you have enough cash to pay off college tuition within the payment period, I stand corrected. Most people don't.
You pay with the card, and then you pay off the card with the student loans. Many colleges don't accept credit cards for tuition, and a local one stopped recently.
Anecdotally, I think at least a few universities have stopped accepting credit cards because of the fees associated with accepting them - the merchant fee on a $30K tuition payment is not trivial. I wouldn't doubt that when offered, most people end up paying with credit cards because of the rewards programs.