when we went to oz we went through america. took about 2 hours longer on the plane but when we landed in San Fransisco we were on Oz time more or less, although ours was an impromptu stop-over i would recommend it, esp as most of the hotels had shuttle buses from the airport and it seemed a nice enough place, we stopped in LA on the way back and again worked well.
I have friends who recently did NZ-London with their 13mo DD. They did like Lou suggests; in fact they even had the quick shower in Singapore too They felt that if their DD was going to be jetlagged anyway they might as just well do the whole lot in one go, rather than having to suffer the pain more than once.
However, if you want to use the opportunity to see a bit of the world with the three stopovers then I would do it!!!
Singapore has a great transit hotel which I usually book for a 6 hr block and then I can go to the pool, eat, exhaust my dd at the play park bit and then shower etc in a nice room. Make the whole flight a much better experience.
I did LDN Melbourne with a 12 months old ds. Did fly with a stop-over and then back without a stop. Oh my god, it was soooo exhausting! I would definitely do a one stop-over. Book always a night flight so baby is sleeping in a cot and you can too or just relax and watch some films... We stopped in Singapore and stayed in Sentosa island for 3 days, stayed by the pool, got over jetlag, relaxed. The time difference between SIN and OZ is not so significant so there was not a real jetlag to get over after that. Enjoy whatever you decide to do and happy return!
I second what zuzkah said. When we moved back to Oz from London we had a 3 day stopover in Singapore and stayed in a lovely hotel on Sentosa Island. It was great to have a mini holiday along the way - lazing by the pool (and spending the proceeds from our flat sale on expensive handbags in the city).
The time difference between Singapore and Melb is only and hour or 2 and the flight is only 8hrs so the last leg is a breeze.
We did this trip when DD was 6mo and again when she was 2.10yo - I was 13wk preg on the second trip and I was the badly behaved-overly-whingy-jetlagged-moany-one!!
We stopped in Penang on the way with our then 11 months old. Was lovely! On reflection it would have been great to have a stop on the way back (just to have some more pool/beach time) and would do that if we ever go back.
I would recommend you have a stopover for a day or two before moving on to Australia having done this journey myself. The second flight to Aus will depart late evening and that'll take around 8 hours.
Going non stop to Australia will mean you'll be on that plane for two days and in economy in particular that can be rough going. The combination of no decent sleep, being served a series of below average airline food and a lack of fresh air can play havoc in their own ways. Also filling in the Aus Immigration forms in the middle of the night and whilst tired is not good!!.