Check in for your flight to Australia with your Australian passport. When you get to the passport control before you board your flight, show your US passport, because that's where your entry stamp is for Singapore, and that's where you need to get an exit stamp.
If, when you check in, the airline asks about your immigration status on Singapore, show them the US passport with the entry stamp. Similarly, if the passport inspector asks about your ability to enter Australia, show the Australian passport. Most countries pay no attention to travelers' destinations at the exit control, but a few do. I don't know about Singapore. In any event, you're allowed to be a dual citizen, so you can show both passports when it's necessary.
(I'm aware of no reason for you not to use the Australian passport for the trip to Singapore; US dual citizens can check in for international flights leaving the US with their non-US passports; I do it all the time and there's never been the slightest hint of a problem. But if you have a good reason for using the US passport to fly to Singapore, follow the advice above.)