(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")
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...")