While you can't use DL miles to upgrade, you can use cash (or NZ airpoints) to bid on standby upgrades (Oneup in NZ branding).
Check the NZ forum thread for details and successful bids, but note you can only use these to upgrade one class of service. So if you have booked coach, you will only be able to upgrade to premium economy.
Last edited by PacificTwo; Apr 10, 2014 at 12:33 pm
Reason: typo