Six continues:
Hold Up + L + R, and press Start at the Konami screen.

Nine continues:
Hold L + R and press Start(5) at the title screen.

Unlimited napalm:
Walk through the first mission without shooting or hitting any targets. Your score should be 0 points. Defeat the Boss without shooting its side gun turrets. Your score should be 2800 points. When the asteroid level begins, press X to change weapons until the unlimited napalm gun is selected. Survive this mission to keep the unlimited napalm for the remainder of the game. Note: It cannot be powered up.

Hint: Failed ending:
Go through the first level, and do not destroy the Boss. Finish the rest of the game to see the failed ending in which everyone dies.