(covers information from several alternate timelines)
A divorce or separation was the official dissolution of a marriage.
In Benjamin Sisko's visions of himself as Benny Russell in 1953, Herbert Rossoff told Kay Eaton he knew "a good divorce lawyer" after she disagreed with her husband Julius' idea for Benny's "Deep Space Nine" stories. (DS9: "Far Beyond the Stars")
After Grilka asked Quark how she could repay him after helping her become leader of her House, the Ferengi asked her for a divorce. Klingon divorce involved knocking him to the ground, saying in Klingonese, "N'Gos tlhogh cha!" ("Our marriage is done!") and spitting on him. (DS9: "The House of Quark")
According to Lidell Ren, "I hope you never have to find out for yourself, but I can tell you, ending a marriage, it's a quiet thing. The arguments are long over and there's really nothing much left to say at all." (VOY: "Ex Post Facto")
In 2374, Larell claimed to have gotten separated from Morn after two years of marriage, after which they still kept in contact often. She said that Quark was a frequent subject of their discussions. (DS9: "Who Mourns for Morn?")
In an alternate timeline, Jake Sisko married Korena, but the marriage ended in divorce after he spent most of his time searching for his father, Benjamin, who had been displaced from time. (DS9: "The Visitor")
In another alternate timeline, Captain Jean-Luc Picard married Dr. Beverly Crusher, a marriage that eventually ended in divorce, although the two remained fairly amiable and Dr. Crusher kept the name "Picard". (TNG: "All Good Things...")