What do people carry to fix flats with clinchers?

I carry all that (minus the cell phone) plus a stick-on patch kit.

I don't ride a bike with clinchers yet (I will be soon).  But what I carry depends on the type of ride, distance, etc.

1.  For short training rides I carry a spare tube, patch kit, small multi use tool, two pry levers, and a pump.  The pump goes on the frame and everything else fits easily in a small bag under my seat.

2.  For races I carry a spare tube, two pry levers and CO2 (all easily fit under the seat in the same small bag).

3.  For long rides (more food to carry and need more space).  I have a larger bag under the seat and an additional frame bag that I carry all of the items in line 1 plus an extra tube or two.

4.  I have seen some people carry these items in water bottles (empty), jersey, shorts, bags, panniers, in a bag attached in a water bottle cage, etc.  Lots of options and everyone has there own preferances.

Michael

Here's a short video on how to fix a flat. Like others said, carry a patch kit, spare tube, couple of plastic tire levers, some CO2, and a pump that works for both hand pumping and CO2 (the former in case I run out of CO2 before I run out of bad luck). I also carry a small multi-tool to handle other mechanical issues that might arise.

I would also recommend taking some time one evening to practice removing the wheel and tire, and "changing" it (just use the same inner tube).  Practice pumping with the frame pump as well as at least once using the CO2 so you have an idea of how they work.  That way, if you are out  15 or 20 miles from home and get a flat (which will be stressfull enough) you won't have the added stress of not knowing what to do.

And don't forget the little lace dollies to put your tools on so they don't come in contact with that nasty old ground.

You've gotten the basic list down from others, but I'd add in some cash.  A $20 and a quarter will get you out of a lot of jams (sidewall tear, bonking and running out of food/water, cab fare, phone call...) All of this goes into an underseat pouch or can be mounted on a rear hydration system like an X-Lab Flatwing.

 If you don't want to carry surgical gloves (or your caught without them) grass will get most of the grease off your hands.  

 And practice.  Use the rear wheel, start and finish with the wheel mounted in the bike and the chain on the cogs.  Use only the tools in your emergency kit.

An empty Gu package also works well for this, I've been told, if you like Gu.

Every ride:

Two tubes
Two CO2 cartridges
CO2 adaptor
Two tire levers
One hand pump
Dollar bill
Cell

A tube, 2 CO2's and inflator, tire wrenches, and a patch kit.  I also carry a small allen wrench set, but not for flats. I use the smallest saddle bag possible underneath my saddle for easy carrying. We sell them. See this link:

http://www.bikesportmichigan.com/shop/product.php?productid=16370&cat=0&page=1

 mrsmdg2003 is still looking for the Grandma's hand made lace doily that used to be under the living room lamp. She'll never think to look in my saddle bag.

Not having read any other replies...

I carry 2 Quick Sticks (the best tire levers I've ever found), a mini pump (Blackburn Airstick) and 2 tubes. The pump is on the frame, and the levers and tubes are in jersey pockets. For races I ditch one tube and stuff the other with the levers into the saddle rails (using a section of old inner tube to tie it in place) and keep the pump on the frame.