Get one of those handy dandy
pet carriers (small version) and strap it to the back rack? Our cat didn't mind riding in hers. Never took her on the bike...the vet was over 30 miles away
I know of one lady whose cat rode in a front wicker basket. IIRC she kept a short leash on it, but the cat didn't seem to mind. The lady (and her cat) were quite the fixture in our area at the time I lived there.
Aaron