One option is the Plaza Premium Lounge in Terminal 2, which offers (unisex) shower facilities for something like S$15, S$40 if you want to use the lounge facilities as well (selection of snack food, non-alcoholic drinks, Internet access). My recollection of the prices may be faulty because I had free access with my Priority Pass membership.

Look for the signs that say "Gym and Lounge" and take the escalators up one level.