Hi there:
There are lots of different ways you can join a playgroup.
1) You could reach out to one of the organized groups (the playgroups at St. George's, for instance) or the New Mothers Support Group (you can google them for the web site). The NMSG can help link you up with other mothers with similarly aged kids. My 16-month-old & I have been a part of a group through the NMSG for the past year.
2) If you live in a condo complex, you could post a sign, suggesting a local playgroup for other kids in your child's age bracket. I did this when we first moved to Singapore, and our local group is still going strong.
3) You could do what a lot of moms have done on this board -- simply start a thread & see if anyone is interested in joining up. Since there are always new people moving to Singapore and/or new mothers eager to meet other moms, you are sure to have some responses.
I've been really lucky in that the two playgroups my daughter & I belong to are filled with interesting, non-competitive, laid back women and fun kids. Hope you find the same! Good luck.