Here is the full list of all 44 Heavy Storm Shadow achievements.
Clear the Desert map once.
Clear the Moon map once.
Clear the Tundra map once.
Achieve one victory at the challenge level of 1.
Achieve one victory at the challenge level of 2.
Achieve one victory at the challenge level of 3.
Achieve one victory at the challenge level of 4.
Achieve one victory at the challenge level of 5.
Achieve one victory at the challenge level of 6.
Achieve one victory at the challenge level of 7.
Achieve one victory at the challenge level of 8.
Achieve one victory at the challenge level of 9 on the Desert.
Achieve one victory at the challenge level of 9 on the Lunar.
Achieve one victory at the challenge level of 9 on the Snowfield.
Unlock 5 primary weapons.
Unlock all mechas.
Upgrade the fire control system to the max.
Upgrade the defense system to the max.
Upgrade the auxiliary system to the max.
Build the affinity level with Alicia, pilot of the Storm Caller, to 5.
Build the affinity level with Cecil, pilot of the Hermes, to 5.
Build the affinity level with Sif, pilot of the Thunder Strike, to 5.
Build the affinity with Rita, pilot of the Obliterate IV, to 5.
Achieve one victory without any airdrop support in single battle.
Achieve one victory without any damage in single battle.
Restores more than 120 HP in single battle.
Increases critical chance of the primary weapon by 100% in single battle.
Achieve victory without BOOST in a single battle.
Achieve one victory without overheating the energy tank.
Slay Giant Spider.
Defeat Meteor Scorpion.
Defeat Desert Scorpion.
Defeat Tortoiselia.
Defeat Giant Umbrella Spider.
Defeat Ice Plague.
Defeat Toxic Stinger.
Defeat Eggsack Spider.
Defeat the Creeping Barbs.
Defeat Scythe Rhino.
Defeat Bone Spikes.
Slay Umbrella Spider.
Defeat Slime Fish.
Slay Faceless.
Beat Wailing Sprout.