I've been living in Singapore for about 18 months without my own car. The trains are terrific and very easy with a stroller (I highly recommend the McLaren strollers). I'm not so keen on the buses as they insist you fold the pram (even the wheelchair buses). There is a terrific baby shop for carseats and strollers on Ballesteir Road called Baby Avenue.
A very popular option for travelling with babies in taxis is the Maxi Cosi Cabrio Fix capsule. You can fix it with just a seatbelt and it is quite safe. Also quick to put in, which is important with a taxi. These capsules aren't legal in Australia, but are quite suitable given how safe the roads are in Singapore. If you do have your own car, there is a base mount that you can install. I have seen these capsules on sale on Ebay in Australia, so you might be able to pick one up before you get here. The Maxi Cosi capsule can be used in conjunction with a variety of strollers/prams. The Quinny option is fabulous, but it's a very expensive pram (at least I think it is). You can also use the McLaren Techno XLR, and a bunch of others. We have the McLaren because you can also attach a skateboard to this one, which our toddler uses.
Once your baby is a bit older, the Joeysafe harness is a really good option. We bought ours directly from the importer in Singapore (Angela,
[email protected], 92776519). It is a padded, full body harness that you put a regular seatbelt through the back of. It is suitable from about 18 months. It's not as safe as a carseat, but it is certainly better than putting a little person in an adult seatbelt, and can be easily stored under a stroller once you get to where you are going.
I hope this is useful. If you'd like to discuss more, send me a PM and we can email or phone. I've been in this situation for a while and know what it's like! Cheers, Katie