Went to the Dublin Yoga Studio in Ranelagh, D.6 (just between Dartmouth Sq and the canal) for a few weeks over Nov/Dec for a beginners pilates class. They do all sorts of classes. www.yogadublin.com and you pay up front for, say, 6 or 8 classes.
Bernie Thompson at the Dublin Yoga Centre, Dublin 12, is very good. She does Ashtanga yoga for sure, and I think some other type. I know it's not the area you're looking for, but if you have a car, it's only a few miles from Dublin 6 and well worth a look. There's parking right outside the door too.
Pilates Classes in Bushy Park (Sportsworld Athletics Club House) in Terenure Monday - Thursday. Course is 6 weeks in duration and caters for beginners and intermediates. Thought they were v.good.
I did find it good, and I'd never done Pilates or Yoga before. Nice girl who took the sessions, very approachable and made sure you didn't push yourself if you were a beginner. It was harder to keep going to the classes with the shorter, dark evenings but now with the longer evenings, it would be easier to keep going...
Yoga Dublin Studios near the Luas station on Charlemont I find good, lots of different classes and worKshops, and stuff on every day, lots of different teachers, all seem pretty experienced...
try Centre Studios in Rathmines Village - close to the Tesco store... there's a hard to spot gateway entrance beside the off-licence and the studio is up a flight of steps.
They do yoga and pilates and have a selection of good teachers