If you are doing good enough in a song in a standard game, a rocket with a spiral tail will appear somewhere during the song. Successfully detonate that rocket, then successfully complete the song to unlock the new firework. Each song on each difficulty has its own unlockable firework, for a total of 45 unlockable firework types.

Firework	How to Unlock
Bee	Sting of the Bumble Bee, Hard
Cannon	Cannon in D, Medium
Flame	Explode to Joy, Hard
Four leaf clover	Cannon in D, Easy
Heart with wings	Eine Kleine Rochtmusik, Medium
Helmet	Sugar High, Hard
Horseshoe	Cannon in D, Hard
Hot Air Balloon	Sting of the Bumble Bee, Medium
Raining Cloud	Sugar High, Medium
Scorpion	Sugar High, Easy
Shark	Eine Kleine Rochtmusik, Easy
Tea Pot	Explode to Joy, Medium
Teddy Bear	Sting of the Bumble Bee, Easy
Top Hat	Explode to Joy, Easy
Windmill	Eine Kleine Rochtmusik, Hard