Singapore Airlines announced the only Trans-Tasman Capital connection service between Canberra and Wellington back in January 2016 (read more here) and started the route in September the same year.
The route pair Singapore – Canberra – Wellington must not have worked too well because Singapore Airlines is shuffling the operations in early May continuing serving these destinations but using different set of flights and connection points.
You can access Singapore Airlines here.
Here’s an excerpt from Routes Online (access here):
Singapore – Melbourne – Wellington eff 03MAY18 4 weekly 777-200ER
SQ247 SIN1945 – 0510+1MEL0700+1 – 1220+1WLG 772 x237
SQ248 WLG1345 – 1530MEL1650 – 2245SIN 772 x134
Singapore – Sydney – Canberra eff 01MAY18 1 daily 777-300ER
SQ288 SIN1030 – 2010SYD2120 – 2220CBR2315 – 0515+1SIN 77W D
Wellington service is four times a week via Melbourne and Canberra’s daily service is triangular via Sydney
So, there is another fifth freedom Trans-Tasman service between Melbourne and Wellington, although Singapore Airlines is not known to trash yields and I doubt that they would do it on this case either.
The triangular service between Singapore – Sydney – Canberra – Singapore is quite interesting. Airlines have cut down on fifth freedoms and have even fewer triangular services left such as this Canberra one left.
As the Sydney – Canberra is a domestic flight, Singapore Airlines cannot sell tickets for that sector alone. You can, however, have a stopover in Sydney on your way to Canberra if flying on SQ.