RLast year I had a return ticket (award) from SQ [MEL-SIN] and a separate paid ticket from LH [SIN-FRA-SOF] and when I arrived at Mel the check-in agent checked my bags direct to SOF saying you are on *A all the way today so there's no problem checking your bags all the way regardless of having two tickets issued by two different companies.
Check-in agent was wrong re star alliance comment, as serfty pointed out this is an (unwritten) SQ policy, and has nothing to do with alliances. SQ will through check to most non-LCC carriers, even when on separate tickets. All of these through-checking policies are carrier specific.