My family of four is hoping to book a trip to Australia next summer using our long accumulated miles on American, in business or 1st Class, hopefully. From reading here I know this isn't going to be easy, and I want to thank many of you here for the excellent suggestions on how to get this done.

One thing about our trip that I haven't seen asked about is stopovers. We are in the Washington DC area, so our options for getting to Australia appear to be through DFW, LAX, or JFK, any of which would be fine. But we were hoping to spend a few days in LA on one end or the other, which leads to my question.

Which direction (US-Australia or Australia-US) is likely to be harder to find flights, and thus the direction that we should be more flexible on our US connection point? Or possibly a different question, which direction is it hard to find a flight through LAX?

Thanks in advance to anyone who can help.