Burnham    
Officer shards can be obtained from completing the following missions: Swings and Roundabouts Part 1 and Part 5. Also, shards can be gained through the Discovery Officer Chest, Ultra Recruit Chest, and exchanging Transporter Patterns. Occasionally, more shards can be acquired through Rotational Ultra Chests.
 
Saru    
Officer shards can be gained through the Discovery Officer Chest and Ultra Recruit Chest. Occasionally, more shards can be acquired through Rotational Ultra Chests and Event Rewards.

Paul Stamets    
Officer shards can be obtained from completing the following missions: Swings and Roundabouts Part 3 and Part 5. Also, shards can be gained through the Discovery Officer Chest and Ultra Recruit Chest. Occasionally, more shards can be acquired through Rotational Ultra Chests. Extra shards can be acquired through the Offers tab.

Georgiou    
Officer shards can be gained through the Discovery Officer Chest. You can also exchange Transporter Patterns to get additional shards.

Tilly    
Officer shards can be obtained from the Discovery Officer Chest.

Tyler    
Officer shards can be gained through the Discovery Officer Chest and Ultra Recruit Chest. Occasionally, more shards can be acquired through Rotational Ultra Chests.

Hugh    
Officer shards can be obtained from the Discovery Officer Chest. Extra shards can be acquired through the Offers tab.

Airiam    
Officer shards can be obtained from the Discovery Officer Chest.