Rowing is probably the only one I can think of down here. Huge with the private schools, unheard of with the public schools. Perhaps some of the other sports where there's a perception of it being a rich person's game (tennis, golf). Most of the other major sports are equally popular in the public and private system.
I went to a Catholic high school which no longer exists (thankfully, I missed that turd Robert Best by one year) - I was on the ten-pin bowling team. No other school would compete with us so during inter-school sport, we'd catch the bus to Box Hill Bowl and bowl a few frames against each other. We were also allowed to skateboard - we were driven in a minivan to Bulleen Bowl. It almost made up for dodging the dodgy Christian Brothers that hung around the place.