I used to carry Winslow the dog in my Zo bag when she was a puppy. I'd put the bag in front of me with her head sticking out. This was only around the city, and not on the open highway.
Once she got a little bigger I stopped.
I knew a girl in here in SF that would just slap her tank and her pit would jump up and just lay accross the seat/tank with front legs over one side and back legs over the other. She took that dog everyplace she went in the city. I don't know how he stayed on the bike.