When I was hour-building, the school I hired from were training someone for his PPL on a Piper Super Cub. He passed his check-ride a week before I left.
I don't recall seeing "Tiger Moth" post on here for a little while, but he's doing his PPL on, you guessed it, a Tiger Moth, and soloed it a little while ago.
Basically, if you want to learn on a specific aircraft, then go for it. As long as it's a single-engine and doesn't have complexities like a variable-pitch prop, it shouldn't cause you any trouble as long as you have an instructor who's familiar with the type. The only things which might be a factor would be cost and availability. But as long as you understand that and go in with your eyes open, I don't see it being an issue.
Good luck - and have fun!
FFF
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