Yup, all of the above: kept them for awhile, then gave away or sold, now only token participate in the games to make my villagers think they have some competition, and then beat me to the trophies!
I'll just keep one of the same trophy until (or if) I get a second one, in which case I'll sell the older trophy and keep the new one. So far the trophy I currently have is in the storage space since it doesn't really fit in with the rest of my house.
I try to keep one of each unique trophy in my trophy room, which I have as a museum exhibit. (I also keep some duplicate villager pics in there). But any extra trophies I get, I usually just sell them.