All modes are available from the mode select screen once unlocked by selecting "Change to Mystery Modes", with the exception of Original Mode.

Mode	How to unlock
Cognito Mode	Complete Puzzle Mode.
Finity Mode	Complete Level 10 in Endless Mode.
Hyper Mode	Complete Level 5 in Action Mode.
Original Mode	When on the regular mode selection screen, rotate the left analog stick clockwise multiple times to start a game in Original Mode.
Twilight Mode	Complete Level 5 in Classic Mode.