My parents' Pom fit on the basket of my old dutch bike. I'd occasionally take him for a ride on the bicycle path, he seemed to quite enjoy it (as he wasn't supposed to do any walking, lol, he's a lazy bum). We (SO & I) recently adopted a stray dog but he's too big to fit in a basket and carry around. He seems to like running alongside our bikes on our street but hasn't quite mastered the technique and usually crosses in front of our bikes scaring the **** out of us. I hope it's just a matter of creating the habit and soon we'll be able to take short trips with him to give him a better exercise than a walk.
I think it depends on your furry friend's level or energy and temper, whether to carry him around or let him run alongside. The Pom is so mellow he wouldn't even move from the basket, never attempted to jump from it (even as a puppy), but my new friend is way too hyperactive to be trust to stay in a basket, even if we found one he fitted into lol.