All venues:
At the main menu, highlight My Corner and press Left, Left, Left, Right, Right, Right, Left, Right, Right.

Big Head mode:
At the main menu, highlight Play Now and press Left, Right, Left, Right, Left, Right, Left.

Small fighter:
At the main menu, highlight Play Now and press Left, Left, Left, Right, Right, Right, A.

Big Tigger:
Select My Corner/Record Books/Most Wins-Boxer, then press Up, Up to unlock the middleweight Big Tigger.

Thanks to Revolution readers Sugar Ray Marv and Tweety!