Blow Me Out level:
Beat all ten planets on normal difficulty to unlock this level in Battle mode.
Pipes and Belts level:
Beat classic mode to unlock this level.
Beat all ten planets on normal difficulty to unlock the classic version of the game with updated graphics.
Searchlights Battle Stage:
Complete Levels 10-5 (make sure you get the key) and you will unlock this level.
Thanks to Revolution Readers rockbomb, racha3l and chri$!