There are many threads on FT about trying to get awards USA Australia
You do not state which class you are trying for: economy, business first.
You need to be very flexible with dates and airports. (LAX SFO - MEL SYD BNE)
Airlines that fly USA/Canada to/from Australia (most from LAX) are
- United (LAX SFO – SYD MEL & GUM-CNS) [Star Alliance]
- AA (LAX SYD) from Dec 2015 [OneWorld Alliance]
- Qantas (LAX JFK DFW HNL – SYD BNE MEL) SFO from Dec 2015 [OneWorld Alliance] (AS partner)
- Jetstar (HNL – SYD BNE)
- Delta (VA & AS partner) (LAX – SYD) [Skyteam Alliance]
- Virgin Australia (DL partner) (LAX – SYD BNE)
- Hawaiian, via Hawaii (AA & VA partner) (HNL – SYD BNE)
- Air NZ, via AKL [Star Alliance]
- Fiji Airways, via Fiji (AA & AS partner)
- Air Tahiti Nui (last segments on codeshares) (AA & DL partner)
- Air Canada [Star Alliance]
And others via Asia & Middle East