Pay bills with a credit card?
Incidentally, this very much includes the case where you believe the service is terminated: maybe, for example, because it’s a utility bill and you moved to a different state a year ago, but they keep charging you anyway. It’s very easy to set this up so companies have access to your credit; it’s next to impossible to get their grubby faces out of the trough if they want to keep drinking there. I’m “once bitten, twice shy” on this one.
I have a friend, one of those folks who buys everything with cash (including cars and houses) who keeps a Discover card just for this purpose. He makes a hundred bucks a year or so on Discover’s cash back. I myself got rid of most of my cards a few years ago and solely use an American Express card, which I use for absolutely everything. I have to pay it off, which enforces some discipline, and using Membership Rewards points has never been a problem for me (as long as you give the process enough lead-time).
As far as getting something back on rewards cards goes, people have already mentioned several options. One thing to keep in mind with frequent flyer miles is that there’s a difference between cards that give you miles themselves and the cards that give you points for airline tickets (these cards often advertise themselves as being good on any airline). Free frequent flyer miles has a very thorough breakdown of rewards cards (it’s probably been mentioned on mefi before). Cards that give you real airline miles usually have annual fees, though you can often get the first year for free. You can check out the offers and decide whether the miles would be more valuable to you than the cash you’d get from a cash back card.