I had a Doberman rescue (as in, she came up on our porch and made herself at home), the late and greatly lamented Vrka, that was great as a biking dog. She was trained to heel, and would run along beside me. Her favorite command was "OK!" which meant that she could run out in front and pull me. I took advantage of her on hills, oddly enough. She would return to heel on command.
That dog, at least, wasn't interested in chasing things while we were going places, she just like to run with her "pack." Great dog. I never had a bit of problem with her on the bike. Other times, well, we won't go into that.
Our current dog, a mini Dashchund, would be a disaster. Ignoring the fact he has 3" legs and couldn't keep up, he'd keep stopping to sniff, chase and threaten everything on the way - probably running under the bike to do it. He stays at home, much to his displeasure.
Give big space to the festive dog that make sport in the roadway. Avoid entanglement with your wheel spoke.
(Sign in Japan)
1978 Raleigh Gran Prix
2003 EZ Sport AX