AA, BA and US are all oneworld carriers, so you will need to look for oneworld routes and award availability.

Use www.oneworld.com to look at the route networks and carriers available.

Some options are:
CX, JL, MH, AA, US for TPAC flights.
CX, JL, MH, QF from Asia to Australia/NZ.
QF (and a few LA flights) between Australia and NZ.
QF from Australia to South Africa
BA, QF, QR, CX, MH, JL for Asia/Australia to Europe.
BA, QR for South Africa to Europe.
AA, AB, BA, IB, US for TATL.

AA and US will be best for the longhaul region-region flights as it has a zone based chart.
BA best for shorter direct flights as it uses a distance based award chart priced per segment.