As a stay-at-home mom, I wasn't too keen on Singapore when I first moved here, but I've grown to like it much more.
One of the things that surprised me (and annoyed me) when I moved here with my husband and 7 1/2 month old was that there's a lot of waiting around (long lines) and it takes so much more time to do things in Singapore than it did in the U.S.
Now, waiting for buses, cabs, trains and elevators might not be a big deal if you're out and about by yourself, but it's a very big deal if you're out with a baby. I would "plan" to be home at such-in-such time to get my child a nap or a meal, but the "plan" usually didn't work. In general, there's no such thing as a "quick trip" to the grocery store or mall in Singapore.
My son was taking 2-3 naps a day when we first got here and getting somewhere when he was awake wasn't easy in such a short amount of time. He didn't sleep well in his stroller. I was often homebound. He's how almost 2 and has been taking 1 nap a day the past few months so it's much easier to do things with him because there's more time.
We also got a car a few months ago and that makes transport much easier with a toddler, although parking can be a challenge. Sometimes it takes a long time to get in and out of garages because 99% of drivers back their cars in and often take several attempts to park perfectly. It might take you 15 minutes to get somewhere by car but an hour to get there by bus and sometimes the MRT isn't going where you need to go.
I don't mean to sound negative, but it was hard for me at first. It got easier once my son got a little older and we got a car.
As far as health care goes, it'll depend on the plan. The one through my husband's company is ok because we're all healthy and don't have any preexisting conditions so it works great for us. But I read over the plan and its title should be the "Don't Get Sick Plan."
Another thing, if you're not getting a car, you might want to consider a car seat that doubles as a stroller because taxicabs don't have carseats. So many people just hold their babies on their laps. We used to clip my son's carseat out of his stroller, collapse the stroller base and put him (while his seat) in the cab that way and put the seatbelt through the designated area on the carseat , so he was in his seat and remained buckled in.
If you find an "indoor playground" you like when he gets a little older, be sure and take him there weekday mornings when schools are in session and avoid weekends and public holidays. I made the mistake of taking my son in the afternoons and it's more crowded and lots of big kids aren't careful around the little ones. Some indoor playgrounds will let kids under are 1 or 2 in for free. They can be a nice change from the hot outdoor weather.
Lots of nice areas to live. I suggest a
condo with a pool, playground, amenities, etc because it'll be easier to entertain your child once he starts walking and climbing. However, you'll have to check to be sure places allow your breed of dog.