You can also fly on BA flight numbers on something like MEL-SYD-SIN or BNE-SYD-SIN and get your BA sectors that way.
Unfortunately I think getting to Singapore this way is going to be more expensive than using QF or other airlines.
But QF don't have a PE offering to SIN from SYD and I've always found SYD-SIN to be comparable (price wise) between BA & SQ and I actually prefer the BA product. Good knowledge about the three way route to get the sectors but I would only ever do that as a last resort as I spend far too much time on planes anyway!