Assault Suit Leynos

Melee attacks are performed using the button. At the loadout screen, unequip any weapons to make sure you don't fire them by mistake when trying for this trophy. Then you can equip extra armor to fill the empty loadout slots. Make sure you have the shield and boost equipped as well.

It is recommended to go for this trophy on stage 5 in Arcade Mode. Most of the enemies there do not shoot at you, so you can find a spot where they land and punch them as soon as they do. Ignore the planes; they may do some damage to the bulkheads but not nearly enough to destroy them. When Lance (the boss) appears, try to maneuver behind him so you're not in the path of his flame shot, then get in a few punches before he turns around. Repeat this until he is destroyed.

You can also get this trophy in stage 2, though it's a little harder as the enemies will be shooting at you. Use the same strategy for Lance, punching him from behind. When Nox appears, concentrate on its large gun while blocking the charged shots with your shield. You must destroy his gun before he reaches the evacuation point at the far right, otherwise you will fail the mission. If you take a lot of damage, make sure to get behind him and away from the gun until your suit energy regenerates. Once the gun is out of the way he is a lot easier to deal with. Just stand right beside him and keep punching until he is destroyed.

The other stages either have airborne enemies that cannot be easily reached with punches, or bosses / time limits that make a melee-only loadout impractical.