I sell them all, because they're weird and creepy and I'm too lazy to transfer any to my storage town!! I still need to collect all the Brewster ones though, I'll save those ones when I get them.
I collect gyroids for the time being, displaying one of each in the museum. I just sell repeats though I might start displaying some in my house soon. At some point, I'll probably have more gyroids than I have museum space, so I'll just keep the ones I like best.
I think they make really annoying sounds, so I refuse to put one in my house. So I sell them. I can imagine the people that have museum rooms full of them!?!? I'd go crazy with all the boinging sounds hahaha