At the title screen in the Classic version, press L, R, L, R, L, R, Right, Select. Touch the door that now appears in level 20 to start the bonus rounds.
Levels 91 to 100 in New version
Collect nine letters throughout the game. You must kill at least four enemies with a single Super Bubble, then collect the key that they drop. After defeating the Boss of that group of levels, it will drop a letter instead of the big gem. You must collect a letter from every Boss in the game (nine total) to unlock the final ten levels.
Play as Lovelun in New version
Complete all 100 New version levels with Robolun to unlock Lovelun.
Play as Robolun in New version
Complete all 100 New version levels to unlock Robolun.
Shoe power-up mode in Classic version
At the title screen in the Classic version, press Select, R, L, Left, Right, R, Select, Right.
Super Bubble Bobble in Classic version
Defeat the Classic version Boss using two players. Then, press Left, R, Left, Select, Left, L, Left, Select at the title screen in the Classic version.