Here is the full list of all 42 Earthfall trophies - there are 23 bronze, 16 silver, 2 gold and 1 platinum.
Unlock all Trophies in Earthfall
Kill 400 Threshers
Kill 400 Whiplashes
Kill 100 Beasts
Kill 100 Blackouts
Kill 50 Blackouts
Kill 50 Beasts
Kill 400 Sappers
Kill 20 Blackouts
Complete Into the Breach on Overwhelming
Complete Extermination on Overwhelming
Kill 100 Whiplashes
Kill 20 Beasts
Kill 100 Threshers
Kill 8000 Drones
Kill 50 Enragers
Complete Radio Silence on Overwhelming
Complete Sunnyvale on Overwhelming
Kill 100 Sappers
Complete Data Recovery on Overwhelming
Kill 50 Threshers
Kill 50 Whiplashes
Complete Breakdown on Overwhelming
Complete Supply Run on Overwhelming
Complete the Extermination mission
Complete all the missions in the Natural Selection campaign
Complete the Into the Breach mission
Complete The Diversion on Overwhelming
Complete The Diversion Mission
Kill 50 Sappers
Complete the R&D mission
Complete R&D on Overwhelming
Complete Revelation on Overwhelming
Complete the Revelation mission
Complete all the missions in the Invasive Species campaign
Complete the Radio Silence mission
Complete the Sunnyvale mission
Complete the Data Recovery mission
Kill 1000 Drones
Complete the Breakdown mission
Kill 500 Drones
Complete the Supply Run mission