Fragments are a powerful new addition to the world of Destiny 2. These items add new effects to your subclass which intermingle with how Stasis works. Unlocking Fragments, at least for the first time, isn’t quite clear. However, once you’ve gained access to Fragments, the process of getting new ones becomes rather straightforward.
How to unlock Fragments
To begin unlocking Fragments in Destiny 2 you must complete Born in Darkness Part 2. This quest is offered by the Exo Stranger after finishing the Beyond Light campaign and completing Born in Darkness Part 1.
As soon as you finish Part 2, the Exo Stranger will offer you Fragment quests. You are limited to buying two quests per week from the Exo Stranger. Right now, there are only six Fragments to unlock, which means you can claim them all in three weeks.
In terms of what these Fragments quests are, players can choose whether they want a quest focused on Crucible, Strikes, or Gambit. Each quest has tasks that can only be completed within that specific playlist. It’s also worth noting that the tasks are different for each class.
For example, the Warlock Vanguard Fragment Quest requires defeats of slowed combatants, frozen combatants, and Dark Reflections collected from Strike bosses. On the other hand, the Hunter Vanguard Fragment requires Dusk Slivers from Rifle final blows, defeats of frozen combatants, and powerful (Majors, Ultras and Champions) to be defeated with Stasis damage.
Right now, Fragments are exclusive to the Stasis subclass. Whether this changes in the future remains to be seen, but it would require a major shakeup of the other subclasses. It’s likely this sort of change would require far too much manpower.
There are several Fragments currently in the game and a few are definitely worth picking up. While you’re pondering the best Fragments to unlock, swing by the Shacknews Destiny 2 guide for even more coverage of this new season and beyond (pun intended).
Sam Chandler posted a new article, How to unlock Fragments - Destiny 2
I don't believe the quests/bounties are class specific. I selected the Vanguard option on Titan and Warlock and both required defeating 10 Champions with Stasis. I would also recommend against abandoning them even if they include chores you don't want to complete.