Here is the full list of all 33 Star Trek: Bridge Crew trophies - there are 15 bronze, 13 silver, 4 gold and 1 platinum.
Obtain all the Trophies.
Rescue at least 120 people in the stand-alone Kobayashi Maru mission
Visit every star system in The Trench
Complete all campaign missions and one of each Ongoing Voyages type
Complete a mission with less than 10% Hull Integrity
Destroy 20 Klingon ships
As the Engineer, maximize engines by routing power from shields and phasers
Perform 5 warps as a Helmsman or Engineer
Complete an Ongoing Voyages mission
Complete a mission as a Vulcan
Finish a mission using the original bridge
Complete a mission as Tactical
Complete a mission as Engineer
Destroy 5 non-Klingon ships
As the Engineer, Repair 15 systems
Play 6 missions with friends online
Disrupt systems 5 times via a system intrusion
As Tactical, scan your first ship
Witness the death of an NPC crew member
Finish a mission without using the shields once
Finish a mission without using torpedoes
Complete a mission as Helm
As the Helmsman or Engineer, perform your first warp in a Campaign or Ongoing Voyage mission
While on a mission, look outside of the spaceship using Space View for 10 seconds
Complete a mission as Human
Complete a mission as Captain
Survive a battle against three or more hostile ships
Successfully transport 30 people
Destroy a ship using only torpedoes
Navigate through an asteroid field unscathed
As Captain, temporarily take over an NPC crewed station during a mission
Answer your first Hail
Complete a task in the training scenario