Hi Helen,
I've been here for just a couple of months now and have found the following helpful:
New Mothers' Support Group
www.nmsg-singapore.com
It's more a Parents group really not just mum's. My son and I went to our first social last friday and the kids there ranged from a few months to a few years old. They also have monthly talks and walks around the Botanic Gardens - which we've yet to go to.
Another good place to go is:
Mother and Child
www.motherandchild.com.sg
which has a drop-in clinic in Tanglin Mall which is run by health visitors. They have other activities too though these seem to be more pre- and post- natal care so may not be appropriate for a 19 month old.
We used to live in a remote village where there were practially no activities for babies and young children so we've also revelled in trying out classes at:
Gymboree (Tanglin Mall)
Little Gym (The Forum)
Kindermusic (Tanglin Mall)
As you are from Sydney, I understand there is an association called ANZA which runs socials including (I think) a play group
www.anza.org.sg.
A SMS says there'll be loads of other good advice elsewhere on this forum.
Happy searching and good luck to you