note that in your situation, the "stops" in HKG and SIN are not "stop-overs", but are in fact "transits". You do not need to clear immigration or customs at those transit points. However, as JetLagged said above, you will still need to show your passport at some security points for identification.
Also as others have suggested, the terminal design at HKG and SIN is different. At HKG, when you leave the aircraft, follow the signs for transit passengers. You will need to show your boarding pass and go through a security screening point, then up the escalator to the departures level where you can find various lounges, including the Qantas Club lounge and several Cathay Pacific/DragonAir lounges. There you can head straight to the departure gate when its time to board. Again you may be required to show your passport for identification purposes before boarding.
At SIN, when you exit the aircraft you are enter directly to the combined arrivals/departure area of the terminal and can follow the signs directly to the lounge. Don't go down the escalators to the arrivals/immigration area. The Qantas Club lounge is up an escalator from the main terminal level. At SIN, each departure gate has its own security screening point. So when heading to the departure gate you will be screened and then will need to show your boarding pass and passport for identification purposes as you enter the gate departure area just prior to boarding the aircraft.
In both cases you should take all your carry-on bags with you. Note that in both cases you cannot transit with liquids/aerosols/gels in excess of 100ml as each of the security checkpoints will confiscate anything exceeding the usual limits for international travel.
However, in SIN, you can purchase duty-free liquids etc and the DF shop will package into a tamper-evident bag and deliver it to the departure gate for you to collect after your security screening and before boarding. But be aware there is a purchase cut-off time for each flight, so if planning to use this duty-free purchasing in SIN (and its generally a lot cheaper than in Australia) then do that before heading to the lounge. The DF shop staff will be able to explain the process to you.
So keep your boarding pass and passport handy throughout the transit process.