As a guide trainer at the AMGA, I have the opportunity to witness what I consider the most common cause of damaged ropes. I think many climbers think that it somehow happens on the way up when the rope is usually slack but what I have had the opportunity to observe many times, is that it almost always happens on the way down while the rope is under tension.
As guides, our greatest concern is usually during lowering if the rope moves laterally over a sharp edge or just rough rock. This happens when we do a poor job of coaching where we want the person to go and they start to fight the fall line of gravity.
For recreational climbers, where lowers are not that common I think it happens during rappels. The person rappelling is usually so focused on what is below that they forget to look above them and notice how the rope is running. Then, as they get lower and need to traverse a bit due to terrain or to reach an anchor, the rope will abruptly move across an edge and bam! That is when it happens. Its actually very common and one that even new guides are not that good at noticing when a situation is likely to happen.
So my advice? Spend as much time looking up as you do looking down.