Here is a video of a sidesaddle class at an Arabian horse show. There are a number of sidesaddle clubs and organizations in the US. Popular especially among people who participate in historic recreation groups, sidesaddle riding is not as insecure as it might look. The right leg is draped over a special hook, to help the rider balance, and a whip is carried in the right hand to replace the cues the right leg would otherwise give. Elegant traditional dress is pretty much required, as you'd look silly riding sidesaddle in breeches.
Update: a new national sidesaddle organization has just been launched in the US - take a look at their site.