I saw an article a while back which explained how a person could travel from Toa Payoh to Clementi for free and all in the comfort of a shuttle bus that would transit through the Grandstand (formerly known as Turf City). Free shuttle services are one of the best kept secrets in Singapore. They take you directly to popular places, and most of them are open to the public.
The list is quite long, so I’ll refer you to another website that does a good job in detailing the various routes and timings.
A few more routes that I find particularly useful (but not listed on the aforementioned website) include the following:
Vivocity to Sentosa – Instead of paying $1 to walk across the causeway or $3 to ride the monorail, there are a couple free shuttle services that give you entry to the island FOC. The pickup point is right outside of the exit near JPot on Level 1 of Vivocity.
Yishun interchange to KTPH – As KTPH is a bit out of the way from the MRT, this shuttle takes you to/from KPTH and the Yishun Bus Interchange, which is opposite of the MRT.
I must mention that the list of free shuttle buses was much longer a couple years back; many services have ceased operations since then. Some people, particularly hospital patients and visitors, find it somewhat annoying that people who do not intend on going to the shuttle’s destination take a seat away from those who are using the shuttle for its intended purpose. So I urge you to use some judgment when using these free services. Perhaps only take them during off-peak hours or when the bus is not at full capacity. The counter argument is that since these shuttles are open to the public and must travel the route anyway, why not take up a seat, give the driver some company, and enjoy the ride?