How about JFK-NRT-AKL-MEL (side trip MEL-SYD-MEL is cheap enough in economy) -SIN-ZRH-LHR-FRA-JFK?

ZRH included for easy transfer and to get SQ's new 77W business class. There is a surcharge for this on RTW. Alternatively half the SQ FRA flights have 77W (and half 744).

Unless you want to stop overnight in FRA the LHR-FRA-JFK will need to be on LH (SQ's flight is too early to make the connection).