You need to break down your search into the segments ie lhr>sin, sin>syd, syd>sin, sin>lhr. There IS availibility and you'll find many F options for two so long as you're prepared to be flexible. Haven't been following C so can't comment on that.
You also need to consider *when* you're flying. Beginning/end of school holidays quickly dry up redemption availability on the legs into Oz - and this isn't always easy to ascertain since the different States have differing school calendars ... usually offset by a week so that can, in effect, make a 3 week "mini-blackout" period.
I faced something similar toward the end of last year, wanting J redemption UK-OZ at the end of September. Plenty of BA availability as far as the US and SE Asia, just couldn't find anything into Oz. Luckily, QF then announced that the long awaited DFW-BNE/SYD flights were going on sale and J redemption opened (aided by an email alert from EF) on this route so was able to book LHR-DFW on BA then DFW-BNE-MEL on QF.
You need to:
1) Get in early with respect to booking award flights - usually close to the 350-ish first availability window
2) Sit with the route maps of the two networks and plan individual segments when through availability is gone
Obviously, if you are set on F redemptions, this becomes harder to source the more QF remove it from their network ... and particularly on those more "exotic" routes away from the prime SYD/MEL-LHR, SYD/MEL-LAX flagship routes.
BD