I took DD on my own to NZ when she was 17 months old on Singapore Airlines. They were fantastic. A few suggestions:
Singapore Airport has a swimming pool. Pack swimming costumes in your carry on luggage. Towels, shampoo etc come as part of the entry price. It was fantastic to be able to stretch and tire out the small person in the 3 hour layover. (Or at least it was until she decided teddy needed a bath.)
Let them know as you are checking if you are a parent travelling alone with a small child. My FIL quietly pointed this out to the staff at Heathrow and they let me check in via the First Class check in counter. No standing in queues.
If you have a beloved teddy/soft toy, ask for a luggage tag for them. Write down the flight numbers and/or a contact address in NZ and attach it to the bear. It's the last thing you want to lose between flights.
DD's a big girl but, at 17 months she still (just) fit in the basinette. However, if your child uses a sleeping bag, take that as well as I found the basinette was far too interesting to get settled in. Bear in mind if there's turbulance you will have to take him out of the basinette and strap him to you for the duration.
Singapore Airlines are fantastic with children. I saw very little of DD on the filght as she seemed to spend most of the time in the galley chatting to the hostesses and being fed tidbits.
If you haven't already, invest in pull up nappies now. Airplane toilet changing tables are not the place to deal with a wiggling baby.