It goes All Day Pleasure, Trail Pleasure, Lite Shod(encompasses ADP and TP, with light weighted shoes), Plantation Shod(Heavy shoes) and Built-up(Pads).
There CAN be padded horses that have not had any issues. My trainer had an ex-padded horse whose legs were normal.. but her teeth required tons of work.
I have only ridden up to lite-shod horses, but their movement, along with pretty, is SO SMOOTH. Granted, he was a racking horse, but.