Forget about pets in cabins, not for long haul anyway, and definitely not international.
Animals must be checked in as "live cargo", and upon confirming with the airline, the cost must be paid when you book your own flight, and the airline must be notified and confirmed within 24 hrs of travel. My cat has flown from Dubai to HK ( George from International Pet Travel was there to meet him and whisk our kitty to quarantine, and our cat was fine. No hassles, no behavioural problems, just wanted food and litterbox by the end of it all.) with no problem with Cathay. The airline will load cat on last and offload first. They also must know so as to set the cargo area for live cargo onto a temp setting of that in the cabin - usually between 22-24 oC, plus dim lights so that your cat feels relaxed. We will also be moving our cat with us to Sweden. I recommend lining the carrier, which will be a good size for the cat to move around comfortably, and lined with absorbent pads, plus old clothing that has been worn by you so that it carries your scent, and relaxes your kitty.
In all honesty, Singapore is not far to fly. I have known people who lived nearby move their four cats and two dogs to the UK with no problem. Yes, two of their cats were the skittish kind...