The quickest thing to do will be to go to the QANTAS website, do a dummy booking and see which aircraft is operating the flight you want. Just click on the flight number and it will show you. Should take less than 2 mins.
I think the quickest would be simply to check the Timetable on qantas.com
You may want to check
http://www.zuji.com.sg/ (the local Travelocity site) for SINBKKSIN which may uncover some only locally available fares.