From Singapore Airlines website on such matters:-
"The use of booster seats is not permitted. However, you can use approved car-type child seats and FAA-approved child harness.
Approved car-type child seats may be used for a child under 3 years but not less than 6 months of age. The seat must be forward-facing and can be secured to the aircraft passenger seat by means of the aircraft safety belt. It must also be approved by a foreign airworthiness authority / government standards:
•United Kingdom (UK) Civil Aviation Authority (CAA)
•Federal Aviation Administration (FAA) of the United States of America as meeting the Technical Standard Order TSO-C100b; or seats that have two markings: "This Restraint is Certified for Use in Motor Vehicles and Aircraft" in red lettering and "This seat conforms to all applicable Federal Motor Vehicle Safety Standards (FMVSS)"
•European Safety Standard requirements of United Nations ECE Regulation 44 (UN/ECE 44)
•Canadian Motor Vehicle Safety Standard (CMVSS) No. 213 entitled "Child Restraint Systems" or CMVSS No. 213.1 entitled "Infant Seating and Restraint Systems"
•Australian/New Zealand design standard AS/NZS 1754 for infant car seats.
Some examples of the labels displayed on child car seat approved under one of the above standards can be found on the right of the page.
FAA-approved child harness can also be used on our flights. However, AmSafe's Child Aviation Restraint System (CAReS) child harness does not fit our Business Class seats on our A380-800 and A340-500 aircraft as well as the First and Business Class seats on our B777-300ER aircraft. You may wish to contact our Singapore Airlines office for more details.
If you are using a child car seat or a child harness, please note that the manufacturing standards and instructions must be provided to our ground staff and cabin crew for reference. The use of child car seat and child harness on board is subject to all child seat safety requirements. Both cannot be installed on seats at the emergency exit rows as well as the rows immediately before and after it".
I would book your child a seat, you really do not want your child on your lap for the flight duration. Also the child will likely be on your lap for both take off and landing.
TBH I would not bother using a car seat for sole use on board the aircraft as it could well be more trouble than it is worth. It may well be too wide for the aircraft seat so it becomes unusable. Also you will have to carry this seat (and these can be heavy) along the aisle of the aircraft and these are narrow even at the best of times.
This may be useful to you as well:-
www.nzta.govt.nz/resources/factsheets/07/child-restraints.html