European classical music

Classical music is a term referring to ecclesiastical and concert music originating in Europe from the 2nd millennium AD whose traditions later spread throughout Earth.

Earth opera is part of classical music. (VOY: "The Swarm", "Virtuoso", "Renaissance Man")

At regular intervals, concerts of classical music were held aboard the USS Enterprise-D, usually in Ten Forward. (TNG: "Sarek", "In Theory", "Lessons")

National Anthems

Parts of the American National Anthem (The Star-Spangled Banner) were heard in TOS: "The Omega Glory" when the American flag was brought in.

Parts of the French National Anthem (La Marseillaise) were heard when the crew was seeing Q's Napoleonic creatures for the first time in TNG: "Hide and Q".

Neither piece was played by characters in the episode or heard by the protagonist but were part of the soundtrack of the episodes, which is why they are presented as italicized background information here.

Unidentified classical music

There are still several pieces of classical music heard in Star Trek that we were not able to identify yet. Here is a listing of those pieces and any information that could be gathered:

The violin melody played by Data as Sherlock Holmes on the holodeck at the beginning of TNG: "Elementary, Dear Data".
The music box melodies played by Rishon Uxbridge's (TNG: "The Survivors") and Janet Brooks' (TNG: "The Loss") music boxes.
The Johannes Brahms piano etude Geordi La Forge plays for a few moments, before he decides to play classical guitar music, shortly before Leah Brahms enters his quarters for the first time in TNG: "Galaxy's Child".
The music played by the woodwind quintet, consisting of Data (oboe), Jenna D'Sora (flute), Keiko O'Brien (clarinet) and two other crew members (playing bassoon and horn) after the teaser of TNG: "In Theory".
The two pieces of classical music Harry Kim plays on his clarinet after the teaser of VOY: "Parturition".
The short piece of classical music Harry Kim plays on his clarinet in VOY: "Tuvix".
The piece of piano music that Kathryn Janeway listens to in her quarters while reading La Vita Nuova in VOY: "Latent Image".