Harry Kim and the Kimtones, featuring The Doctor
Jazz was a style of music and/or dance that originated in New Orleans, USA, on Earth in the 20th century. It was characterized by a flexible and strong rhythm. This musical genre was known for its improvisatory content. One of the styles of jazz was Dixieland. (TNG: "11001001")
One night in 2150, T'Pol left the Vulcan Compound and strolled the streets of San Francisco. Attracted to "unusual, chaotic" jazz music, she entered the Fusion night club and listened to the saxophonist. The experience elicited an emotional response she never forgot. (ENT: "Fusion")
Beverly Crusher studied jazz dance in St. Louis. (TNG: "Data's Day")
William Riker was a jazz player. (TNG: "Remember Me") His primary musical instrument was the trombone; however, his skills were not up to that of a professional. (TNG: "11001001") He had trouble with the song Nightbird. (TNG: "Second Chances") Riker could also play the piano, and taught Amarie, a musician in a bar on Qualor II, some jazz routines when the USS Enterprise-D visited the planet in 2368. (TNG: "Unification II")
When Geordi La Forge wanted to impress Doctor Leah Brahms who was about to visit him in his quarters, he decided to play a piece of soft jazz, before finally deciding on a piece played by a classical guitar. (TNG: "Galaxy's Child")
Odo mentioned jazz in 2369, in a hypothetical situation as to why he had never dated anyone. He told Quark that a compromise could lead to someone "spending an agonizing night listening to Klingon opera" instead of watching karo-net or at least listening to jazz. (DS9: "A Man Alone")
Harry Kim was also a jazz musician. He had a band on USS Voyager called "Harry Kim and the Kimtones". The band was an instrumental jazz quartet fronted by Kim. In 2376 they put on a show for the Qomar, who were visiting the ship. The Qomar did not receive them well and they were forced to give the spotlight over to The Doctor. (VOY: "Virtuoso")
In 2375, the Ba'Neth damaged Tuvok's mind with an energy weapon, causing him to lose much of his intellect and to become emotional for a time. While he was in this state he was found to be fond of jazz. (VOY: "Riddles")