Australia and NZ have a separate Chappell/Hadlee trophy for their one day contests. Slightly ironic after Trevor and Greg's contribution. Presumably NZ have held that one from time to time.Frank Worrel and Trans-Tasman trophies respectively.
New Zealand only have a named trophy for that fixture, which perhaps impedes their performance as it’s the one trophy they can’t win.
Pakistan the side with no trophies to play for as far as I can find.
there’s a list here:https://en.m.wikipedia.org/wiki/Test_cricket