Expulsion was the process whereby a student's enrollment in a school was revoked.
James T. Kirk's classmate R.M. Merik was dropped from Starfleet Academy for failing the psychosimulator test. (TOS: "Bread and Circuses")
In 2368, Starfleet Academy Cadet Nicholas Locarno was expelled due to Nova Squadron's fatal failed execution of the banned Kolvoord Starburst maneuver. When Wesley Crusher opined that all the members of Nova Squadron should have been likewise expelled, Jean-Luc Picard informed him that were it not for Locarno's taking full responsibility for the act and pleading on their behalf, they would have been. (TNG: "The First Duty")
In 2370, during a conversation with former Squadron member Sito Jaxa, who had since graduated and been assigned to USS Enterprise-D, Picard stated his belief that Sito should have been expelled for her actions. (TNG: "Lower Decks")
Tom Paris was expelled from Starfleet for causing the deaths of three officers due to pilot error. (VOY: "Caretaker")