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