Join Date: Oct 2004
Location: MUC/LAX/SMV
Programs: LH, UA, BD, AA, NW, FB, NH, AC, Sixt, Hertz, Avis, *W, HH, Marriott, PC, Leaders Club, AMEX
Posts: 12,406
I just checked the system and is shows me free award seats in F (O) and C (I) on SQ325 on March 23, and also on NZ82, your connecting flight.
Alternatively, FRA-HKG-AKL on LH738 and NZ86 is available, too.
And yes, LH776 is open in O and I, as well.
Btw, SQ25 is available, too, so you could spend the day in SIN, visiting the city, and then continue to AKL in the evening.