Most are found in static places throughout the world.

In subsequent playthroughs, all of the rings from previous playthroughs are available as well, so you can't miss out on anything that you didn't get by going to the next playthrough. The +2 versions (and single +3 version) can be found in playthrough 3 (again, in unique locations). The +1 versions can be found in playthrough 2 (in unique locations).

The basic versions of each ring can be found in your first playthrough.