Chopper Attack

Level select:
When you see 'Press Start' at the title screen, press C-Up 7 times.

Debug mode:
When you see 'Press Start' at the title screen, h Z and press Right, Left, Up, Down, A, B, Start

King 10:
Locate King 10 in level 7, defeat him, then complete the mission. 

Stingray:
Locate Stingray in level 7, defeat him, then complete the mission. 

Shoot down the President:
Press Z + C-Up + C-Down during gameplay, then fire a homing cluster at an enemy plane.

Power chopper:
Hold L + R + Z and press Start at the main menu before selecting a chopper. 

Plasma weapon:
You can get the alien homing plasma weapon by completing level 7 on expert level and then fighting on the moon.