I shot rimfire bullseye with my father growing up, but I live in a city now and have gotten away from the hobby. Action shooting takes up all the air in American gun culture; I always found it very calming.
Action shooting is definitely the hotter sport right now. But I think bullseye will stay around, especially as the current generation of action shooters ages into it. It has one huge advantage over action: every competitor can shoot at the same time, up to the capacity of the range. If you go to a bullseye match, you get to shoot for a few hours. If you go to an action pistol match you spend hours waiting around only to shoot for a few minutes.