As noted award in F are notoriously difficult to find and I have never seen a pattern, though you can generally forget finding anything during peak travel seasons and school holidays in Australia.

In the last two years, I don't think I've ever actually seen anything come up on the SYD/LAX route. A few here and there very randomly on MEL/LAX (but no pattern, though it seems to generally pop up during less popular months 3-5 months out). The most availability I've ever seen has been on the SYD/DFW route.

JFK/LAX in J on QF can also be hard to come by, but not impossible. However, if willing to compromise (especially if you do find a seat in F onwards to Australia), AS generally does have seats available.

Best advice really is to be diligent in checking daily and ultimately being extraordinarily flexible with travel dates. If you see it, book it, then think through it later.