Please keep the animated SVG's away! Motion is extremely effective at drawing the eye's attention, and so should never be used unless that's the goal. In fact I can't think of a single good design reason to ever animate an icon except for loading/progress icons, which already are ubiquitous and generally don't need any dedicated animation tools since they're so simple and reusable. You don't ever want normal interfac…
I can think of a few cases where animated icons make sense.
1. Control feedback. If your icon deforms slightly under your mouse when you click, it provides extra feedback indicating that your action was registered.
2. Notifications. As you, yourself said, motion catches attention. So animated icons would serve well for any kind of event notification. New message or event, action completion etc.
3. Progress indication. An icon for something that takes a long time, like a file download, could be animated to indicate that the process is still ongoing.
I am sure there are other examples.