Dog fights are usually defined by the merge; the point at which two aircraft pass head to head and then try to get on each other's tail in order to get a radar lock or guns kill.

There is an ideal entry speed at the merge called the corner speed. For example, an F-16's corner speed at around 20,000 ft is around 400-440 knots.

At this speed the pilot can pull maximum instantaneous G (9Gs) and get the best turn rate whilst bleeding the least amount of speed (energy). All things considered equal, this gives him the best chance of getting his nose on the target and getting a kill.

The speed of sound at 20,000 feet will be over ~650 knots. You're going to have a much slower turn rate (and make a much bigger circle in the sky) at that speed as you're still always limited to 9Gs.

Thus, you wouldn't ever enter a merge supersonic if you wanted to engage in a dog fight. You'd go supersonic in order to expedite the intercept (which you would convert into a merge if you couldn't fire BvR), but you would aim to be at corner speed as soon as you got into knife fight (dog fight) territory.

In terms of modern fighter aircraft capable of reaching supersonic speeds engaging each other in battle, there are a few examples. The details of the engagements are tough to find because of the lack of witnesses, but as far as we know 4th generation fighter aircraft (capable of supersonic speed) faced off in the Eritrean-Ethiopian War, the Kosovo War, the Gulf War, the Lebanon War, and, potentially, the Iran-Iraq war.