I've fallen three times on the road I believe. Once because I looked back and stopped because my camera batteries fell out. Looking back caused me to tip over the wrong way.

The other time I was biking with a friend who was going really slow up a hill as was I and I fell over because I couldn't keep my balance. That was before I got really good at track standing.

I know there is a third time, but I cannot remember.

I've fallen many many times while mountain biking. Though I'm using road pedals and shoes on my mountain bike so I'm guessing that doesn't really help.