Master Cheat Mode
Go to the main menu and HOLD L and press Right, Right, Left, Left, Down, Up, B. This unlocks EVERYTHING!
Get Slow Deaths
Go to the main menu and press Up, Up, Down, Down, X, Y, Y, X, Up, Up, Down, Down, X, Y, Y, X
Toggle Infinite Supers
Go to the main menu and press Up, Up, Down, Down, A, X, A, X
Toggle AI
Go to the main menu and press Up, Up, Down, Down, A, A, B, B, X, X, Y, Y
Toggle One Hit Deaths
Go to the main menu and press Up, Up, Down, Down, X, Y, Y, X
Get Dark Phoenix
Beat Arcade Mode using Phoenix
Get Bishop
Beat Arcade Mode using Gambit
Get Psylocke
Beat Arcade Mode using Betsy
Get Bastion
Finish Story Mode using Magneto without losing any matches. You should fight Bastion. Defeat him to unlock him (keep in mind you only have one chance if lose you will have to do the whole story over)
Get Bishop
Beat Arcade Mode using Gambit
Get Blob
Beat Arcade Mode using Bishop
Get Sentinel A
Beat Arcade Mode using Cyclops
Get Character's 3rd Costume
Beat Story mode using any character
Get Character's 6th Costume
Beat 10 characters in time attack mode using any character in under 15 minutes
Get Character's 7th Costume
Use team mode and fight with 8 characters. Defeat the entire opponent team by only using 2 or more of your characters to unlock the 7th costume for the characters you used:

Wolverine gets his original yellow/blue/black costume.

Phoenix gets a red/gold costume similar to Dark Phoenix.

Juggernaut, Gambit, and Psylocke get a costume similar to their comic book original.

Beast gets a blue costume with a yellow X on his chest, and he also has a helmet.
Various End Sequences

Using story mode can result in these different endings

Magneto Victory Ending - Defeat Bastion using Magneto

Phoenix Victory Ending - If you lose the Magneto vs Bastion fight you can fight Phoenix vs Bastion

Secret Ending - Get a PERFECT victory in the Phoenix vs Bastion fight and you will unlock Dark Phoenix