Right now the best availability to/from Australia appears to be on Virgin Australia which is accessible with Delta miles. This is for both business and economy.

You need to search specifically for MEL-LAX / LAX-MEL on their website - it tends to choke up when you don't use LAX as an origin or destination and prices the flights too high. You then need to click dates on the calendar, they won't show as 'low' price seats on the Delta calendar.

For business class...Qantas via AA is very challenging.

United is decent for economy class availability, not so much for business, but better than Qantas via AA.