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