Finally, your Artifact Slot will only unlock after taking on another Vault quest called Cold as the Grave, which is the 16th main mission.Your Fourth Weapon Slot will unlock after completing the first vault mission in the main story, Beneath the Meridian, which is the 10th main mission.Your Class Mod Slot will unlock after helping Rhys out on Promethea and defeating the boss and completing the quest Hostile Takeover, which is your 6th main mission.Your Third Weapon Slot will unlock after leaving Pandora for the first time, completing the Taking Flight and Sanctuary quests at are your 4th and 5th main missions.
These unlocks are tied to completing story quests, so here's when you can expect to unlock them.