I log my rides with Cyclemeter. As Seattle Forrest suggests, I keep it in my pocket--I'm not using it to monitor my progress so much as to record it; it can also give spoken updates at regular intervals if you want. Free to use, but I pay for a "premium" subscription.

There are secure stem mounts for iPhones--check out Quadlock's products for starters.

The potential downside to tracking your rides strictly with GPS is what I call "GPS freakouts" where your device loses its GPS lock and you discover when you review your ride that you hit 700 mph for a brief period. iPhones have gotten a lot better about maintaining GPS locks, and Cyclemeter will keep track of how long your rode with a poor GPS signal. The real solution to this, if it matters to you, is to get a bluetooth wheel-rotation counter, which Cyclemeter (and presumably other apps) can hook into, along with HRMs and the like. iPhones cannot pick up on ANT+ signals. Wahoo used to make an adapter, back in the days of 30-pin iPhones, but most of these performance trackers emit bluetooth anyhow now.