Invincibility:
At the Press Start screen, hold L + R and press Left, Left, Right, Down Down, Up, Right.

Level Select:
At the Press Start screen, hold L + R and press Left, Down, Down, Right, Right, Left, Down.

All Items:
At the Press Start screen, hold L + R and press Left, Left, Down, Right, Right, Up, Left.

Thanks to Revolution readers Maximillian, forces34 and Mark Henry!