I am flying from Los Angeles to either Sydney or Melbourne using frequent flyer points for business class.

I am flying from Los Angeles to either Sydney or Melbourne using frequent flyer points for business class. Are there days of the week that are excluded, doesn't seem to be anything offered Thurs-Sat.

What airline?

It seems you are looking to fly.

Assuming Qantas - seats generally go on sale 353 days out and Business seats get snapped up pretty quickly

Have you considered Dallas Fort Worth DFW - how soon is your requirement? I agree with moa999, you need to get in months ahead of time.

Warning to punters: Contrary to what you may have read on the interweb, redeeming points for premium class travel is not as effortless as you may have been led to believe. Be fully cognisant as you purchase miles/points for that intended redemption that premium class redemption to the U.S. is hard to come by. Premium class redemption to Europe is relatively easier if you are flexible and do not mind flying carriers other than Qantas.

Try Brisbane to Los Angeles,good luck and have a nice trip

Assuming Oneworld type redemption.  Have you looked at going the long way via HKG with CX?

To utilize those points you must be supermegaflexible in dates, route and airline selection. And yes, it almost useless to look less then 6 months before flight.

Presuming you are flying Qantas, mm:

Generally QF prices it's retail SALE fares (lower yield) for Mon-Thur travel on the US routes. Hence, travel on Fri-Sun commands a higher price (usually by $50-75 one way) and attracts the higher yield fare travellers. These higher yield fare travellers are competing for more of the upgrade options (and will likely be higher status FF members), which is why you don't see any or extremely limited availability in the classes above Economy.

There is generally no 'embargo' on QF Australia-USA or v.v. on any particular day - it all comes down to availability. Try to avoid School holidays (both Australian and US) or other sigificant event / drawcard dates. And travelling as a single is always easier than trying to snag multiple seats for partner / family.

Given you are looking to upgrade / redeem in Business, you are possibly better off to look at trying to source available Business seats on the Mon-Thur part of the week. As noted in posts above, available Business seat redemptions to the USA are notoriously hard to nab at anytime unless you are Plat1 or Platinum OR are booking 12 months in advance of your preferred travel.

That said, some interesting routes are sometimes available if you look hard enough. A few months ago, I was presented online with the option of a SYD-SCL (Santiago)-DFW trip at roughly the same redemption rate as SYD-DFW non-stop. Keep your dates flexible as possible and look at all the alternatives. As @11sjw suggests, a LAX-HKG (CX)-SYD (QF) might be long .. but indeed enjoyable.

The best bet would be with VA. They are the most generous and do release seats on weekends. The only problem is VA points aren't easy to come by, you can't simply purchase miles. They also don't have that many partners to compete for the seat, hence the availability is a lot better. 

We are actually quite lucky in this regard as we have the easiest ways to earn VA points. People from other continents can't even join velocity program! 

Thank you so much for your generous adivce, yes it is with Qantas and I am booking for 353 days ahead.

And do not give up if you found nothing - you have to monitor QF website daily m- seats came and go.