Choose the Cheat option at the Setup menu and enter the following:

All Boxers
Start, A, Y, X, X, Z, Z, X, Y, A, Start

Invincibility
Start, A, A, Y, Y, Z, Z, X, X, Start

Intercontinental Tourney
Start, A, A, A, Y, Y, Y, X, X, X, Start

Conversation Mode
Start, Z, A, Y, X, Z, Z, Z, Start

Boxers Life Mode
Start, A, X, Y, Z, A, X, Y, Z, Start

Scrap Yard Arena
Start, Y, Z, Y, Z, A, A, Start

New Outfits
Start, A, Z, Y, X, Start

Turbo Mode
Start, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Start.

Rapid Power Up Mode
Start, A, Y, A, Y, A, Y, X, X, X, Start

Thanks to Revolution reader Adam Kirch!