A recent award redemption on NZ did not charge me any fuel surcharges. Just had to cough up US Sept. 11 fee, Australia "safety and security" charge, and Australia passenger service charge. $30.80 total bill for a one way business class ticket.

My paid return ticket had $250 in fuel surcharges.

Generally speaking, I find awards on MileagePlus/OnePass to be the best in the industry, and OnePass rewards are easy to book. The closest competition is probably US Airways, albeit only for partner flights.