I see a few options, but the problem is, they might not earn very many points from these tickets at all (see my comments at the bottom of this post!).
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(1) Cathay (CX) Asia Miles plus Velocity.
CX because:
-Can earn for the Qantas flights, obviously
-Lufthansa (LH) has just signed on as a points earning partner with CX's program (BUT see my comments below before you get too excited haha...):
https://www.asiamiles.com/am/en/about/news/detail?refID=f6c15d567a4fa510VgnVCM62000007d21c39RCRD
Velocity because:
-Family pooling or transfer possible:
https://www.velocityfrequentflyer.com/content/ProgramBenefits/FamilyBenefits/
-From what I've seen, even the cheapest Singapore (SQ) fares will earn 10% mileage, which isn't much...but better than nothing I guess:
https://www.velocityfrequentflyer.com/content/Earn/Airlines/PointsEarn/#singaporeairlines
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(2) There IS one program to which all three of these airlines (LH, SQ, CX) are eligible points earning partners: Air New Zealand.
And NZ is also a partner to Virgin Australia, so points can be redeemed for Aussie domestic flights on VA.
https://www.airnewzealand.com.au/earning-on-flights
But again, see my comments below...
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(3) If you're aiming to transfer points into your own accounts, you could credit:
-CX segments to Qantas and transfer everything to your own account:
https://www.qantas.com/fflyer/dyn/program/usingPoints/familyTransfers
-SQ segments to Velocity and transfer everything to your account
-(In this scenario, I can't think of where the LH points could go, though...)
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**Some important warnings and additional comments!**
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Because you bought the tickets based on cost (so I assume they are the cheapest discount economy fares), the points-earning from each segment of the journey may not be very much - or may even be zero. Check the booking classes/fare codes of each segment closely. If any of the segments are codeshares, you may even have to call the operating airline to find out which exact fare code will be used to calculate points earning for that particular flight. (Hope I'm making sense!)
-Further to the above^: be warned that LH and SQ are very, VERY stingy when it comes to points earning on their cheapest economy fares, even on their own programs.
-I would advise against signing them up to LH or SQ's programs. These programs aren't a great choice for those who want to make Australian domestic redemptions (it sounds like that's what you're thinking for your relatives?).
-Points transfers may or may not be possible; I think most airlines have a minimum number of miles for making a transfer. Double check.
-CX Asia Miles are only valid 3 years after earning, so if your relatives don't anticipate flying/earning many points within 3 years after this trip, it may not be a good choice:
https://www.asiamiles.com/am/en/abo...RCRD#975c378dc48ea110VgnVCM1000000ad21c39____