Shakespeare has inspired numerous musical works.
Several of his tragedies and a few comedies, including ROMEO AND JULIET and OTHELLO have been turned into operas and ballets. For example, Verdi wrote OTELLO. and FALSTAFF, You probably know Tchaivovsky's theme for the ballet ROMEO AND JULIET.
Felix Mendelssohn composed incidental music for A MIDSUMMER NIGHT'S DREAM, including a rather famous wedding march.
ROMEO AND JULIET was also the basis for the musical WEST SIDE STORY, although no songs specifically mention Shakespeare or the play. A few other plays have been made into musicals, including TWO GENTLEMEN OF VERONA and COMEDY OF ERRORS (which became THE BOYS FROM SYRACUSE).
KISS ME KATE by Cole Porter is a musical about a musical version of THE TAMING OF THE SHREW. This is the show that has the great number "Brush Up Your Shakespeare." ("They'll think you're a helluva of a fellow/If you quote a few lines from OTHELLO")
Other songs:
"Just Like Romeo and Juliet" - this song was originally done by a band called the Reflections, but has been covered a couple of other times. My favorite version is by Scott Kempner.
"Romeo's Tune" by Steve Forbert was actually a hit in 1979.
"(Don't Fear) The Reaper" by Blue Oyster Cult was a hit song about making a suicide pact with your love. ("Romeo and Juliet are together through eternity/We can be like they are")
Lou Reed's "Legendary Hearts" makes a joke of "Romeo, Romeo Wherefore art thou Romeo?" Unfortunately, he makes the same mistakes a LOT of people make, and he mixes up the word "wherefore" (which means "why") with "where."
Speaking of Lou Reed, his brilliant album NEW YORK opens with the song "Romeo Had Juliette," which is a sort of an updating of ROMEO AND JULIET set in contemporary NYC.
"Prince Hal's Dirge" by Loudon Wainwright III is based on the two HENRY IV plays.
"Hamlet Meets John Doe" by the Rave-Ups is a very cool song, even though I don't understand half the lyrics.
"Suicide Is Painless," the title song from the movie M*A*S*H, makes an allusion to Hamlet's "To be or not to be" soliloquy.
Kid Creole and the Coconuts interpolated Shakespeare's "Sonnet 116" ("Let me not to the marriage of true minds admit impediments..." into their song "Agony...Ecstasy."
Bob Dylan frequently throws in allusions or direct references to Shakespeare and Shakespearean characters. For example, both Ophelia and Romeo show up in "Desolation Row" and Will himself is mentioned in "Stuck Inside of Mobile with the Memphis Blues Again."
Finally, the soulful singer Terry Evans has a song called "Shakespeare Didn't Quote That."