Thanks inland ... we use a GoPro Wrist Hero
camera. Not the best quality and the camera itself is definitely spartan but very reliable. They did come out with an HD version of the camera since I bought mine. We've also got a Panasonic SDR-SW20 but our first one leaked water so I'm hesitant to use the "improved" version Panasonic sent me for fear of bricking that one too.
The Lumix is a point-and-shoot, yeah? We've also got a Pentax Optio W80 that we use as a underwater point-and-shoot.
As for securing it, I jury rigged a urethane camera case attaching an athletic urethane strap through the belt loop allowing me to secure it to my arm. I picked up some sort of lanyard from a sporting goods store to tether the camera (through the wrist loop) to the case. It can be a little cumbersome but it works and I've caught waves with it ... I'll even bust it out while on the wave lol.