There's a list of indoor playgrounds suitable for babies under the age of 1. http://bumblebeemum.net/2015/02/23/best-indoor-playgrounds-for-babies-and-toddlers/ - Hokey Pokey @ Seletar Mall - Royce Kids Gym @ Liang Court - Tickle Tickle @ Delta Sports Complex
Petite Park at Changi City Pt on weekdays is usually less crowded but do watch out for the occasional older kids who will run freely, although the staff will tell them not to, for safety reasons.
In addition to that, kiddy fun at Grandstand is fun for young babies too.
thanks