Beat the space faerie in the battledome. To get her as a challenger, you need to redeem a rare item code.
Beat The Black Petri in the battledome.
Beat Kasuki Lu in the battledome. look up his collectable card to get him as a challenger.
Beat Meuka in the battledome. To get Meuka as a challenger, your pet must have Sneezles, then visit the Quick Reference.
Beat Ghost Lupe in the battledome.
Fight with or against a jetsam in the battledome. The results does not matter.
Win a 1-player battledome game (against ANY challenger) and refresh at the winner's page, the page where the trophy is shown. Needs quite some bit of refreshing to get the avatar.
Fight with, or against a Darigan Tonu in the battledome. The results does not matter.
Fight using a Shoyru and win any battle in the battledome.
There are two ways to get this avatar:
1) Fight Balthazar (not Inflatable Balthazar!) in the one player battledome and lose.
2) Lose any battle using a Lupe.