Flat Finger most of the time. Classes I generally play (Engie, Medic, Pyro) don't require precise movements. In fact, they reward broad movements back and forth.
It's actually a mix of Flat Finger and Palm, with a bit of Claw, I guess. Most movements are done with the forearm, unless I'm wrenching something whereupon I rest my hand on the table. If I'm playing and need to switch to precision aiming (Spy, for example), I use the Claw to aim.
I've got a Microsoft Sidewinder mouse. It's too expensive, but if money is no object I highly recommend it. The mouse buttons have a satisfying click, but are easy to press for precision... preciseness. It's got a nice, chunky scroll wheel, and two vertical side buttons that are awesome for extra features. They're right in range of the tip of the thumb, so I'm trying to set up some bind scripts for them.
It's got a good laser system with three mode select buttons: 400 dpi, 800 dpi, and 2000 dpi. The mode switch buttons are right under the scrollwheel and easy to use. So you can swap out your DPI depending on the job.
Finally, it's got a weight system. I crammed it to capacity, my mouse is now a brick at 30+ grams. This is good, as my control style lends to wide sweeping arcs by the forearms that tend to launch lighter mouses.