I spent several months last year stalking QF's inventory in preparation of booking a reward trip this coming November, using a combination of QF's website and EF to check daily redemption loads and existing levels. The trick is to be flexible on your dates and destination, when planning we had a 6 week window in which to book a 3 week vacation, not always realistic for most, but definitely helpful. In the end we wound up booking the following trip using AS miles:

FJ: LAX>Fiji (stopover for 4 days) - 2 in J
FJ: Fiji>AKL - 2 in J

Coming back:
QF: CHC>SYD (stopover for 5 days) - 2 in J
QF: SYD>MEL (23 hour layover) - 2 in J
QF: MEL>LAX - 2 in F

When I booked the outbound I called AS and reserved the LAX>Fiji leg the very first day possible, then 4 days later when award bookings opened up for the AKL leg I called them up and added that leg, creating the stopover in Fiji. It was a 5 minute call and the AS rep was great to work with.

For the return I snagged the MEL>LAX leg online (set an alarm for the middle of the night), then called later that day and added the originating flight in CHC going to SYD for a stopover, then onto MEL for the layover.