I don't know what other people use, but I've used an Olympus Tough point and shoot still camera which also takes video. I wasn't happy with the quality of the video, so I bought a 1080p HD handheld that is surprisingly good and also waterproof to 10 feet which is all you ever need on a kayak or AI/TI.
The one I got was a Kodak Playsport ZX3. It's about the size of a cell phone and weighs about the same too. It's not for everyone, but if you just want to take clips of things, it's pretty good. Other then being waterproof, I like that it's capable of 1080p video, but also up to 60 frames per second in 720p, meaning you can get good high quality shots of action, as the frame rate is fast enough to catch it all. It also takes still pictures at 5 megapixels, which is often enough for most people.
The best part was that the camera only cost me $150 Canadian at our local Best Buy (our dollar is close to par with the U.S. Dollar right now) That's cheap for an entry level 1080p camera that's also waterproof.
