Yes tere are lots allowed on the road but the problem is that you most likely do not have a licence to drive one on the road unless you have full car licence or passed a full motorcycle test prior to 1st January 1997.
Quads and trikes are treated as cars and fall into category B1 which exists for those that passed to bike test prior to the date above. For anyone who did not then B1 does not exist and they fall into the same category for licence purposes as a car
What would be the point of getting a quad bike?
The point of commuting using a scooter is that you can filter through the traffic, and move to the front of stationary traffic stuck at lights.
If you were in a quad, you wouldn't be able to do that, so you'd get cold and wet for absolutely no reason at all.
Whether you can legally drive a quad in traffic depends on where you are. They are illegal in most states, but can be licensed for the road in a few. In Mexico, you often see them carrying a few schoolgirls around town.
Why would you want to be seen on one, however? Quads are for chimps.
