While I've never done it myself, I've known others who have.

Strangest one I ever saw was in an Vetter Windjammer fairing. Guy had the mamma cat and her kittens. Litterbox on one side, food/water/nest on the other. Interesting.

I've seen various boxes on the back of bikes for various dogs. Largest was a crate like thing in place of a saddlebag, where a ~30 pound dog would ride. I've also seen plexiglass and/or wire metal cages either in place of a trunk, or on top of a trunk for smaller dogs and cats.