Two additional ideas away from the central city:
Ride the MRT subway (mostly aerial) for a circular tour of the island, past some of the most sustained apartment tower block developments you'll ever see. Get off at something like Chep Chang Mai to ride the driverless aerial LRT past further megadevelopments at close range. This is the real Singapore where 90% of the inhabitants live.
Visit Changi Prison Museum, near the airport. Prison used by Japan as a brutal POW concentration camp during WW2, now modernized and the site of occasional whippings and canings.