A meteor shower (also called a meteor swarm, meteor storm, or asteroid shower) is an astronomical phenomenon caused by meteors that posed a hazard to starships.
In 2266, Harcourt Fenton Mudd accused Captain James T. Kirk of exceeding his authority when he pursued Mudd's ship into an asteroid shower. (TOS: "Mudd's Women")
In 2269, the USS Enterprise discovered Winston's trading vessel adrift in the Romulan Neutral Zone, an apparent victim of a meteor swarm. (TAS: "The Survivor")
In 2270, Spock noted that the historical records of the Dramen star system were "spotty, at best, going back nineteen years," reporting that "mostly, they're technical data, meteor showers, auroras, [and] radiation reports." (TAS: "Albatross")
After the USS Excelsior was hit by a subspace shock wave created by the destruction of the Klingon moon Praxis in 2293, Captain Hikaru Sulu's first comment after the ship was safely clear of the shock wave was "that was no meteor shower." (Star Trek VI: The Undiscovered Country; VOY: "Flashback")
In the early 24th century, a shuttle piloted by Jean-Luc Picard and Jack R. Crusher was heavily damaged by a micrometeoroid shower while returning to the USS Stargazer from an unauthorized excursion to Argelius IV. (PIC: "No Win Scenario")
In 2370, Cardassian Professor Natima Lang claimed that her Cardassian shuttle was caught and damaged in a meteor storm. It was later revealed that the damage caused to her ship had actually came as a result of Cardassian disruptor fire. (DS9: "Profit and Loss")
In 2376, Seven of Nine enhanced an iso-modulator to correct the Delta Flyer's hull ablation in the event a meteoroid stream was encountered. (VOY: "Tsunkatse")