There are 20 musical references in the script for Julius Caesar, 17 of which are diagetic stage directions:
Most common in Shakespeare's histories, the term alarum signals danger (an alarm), especially in the context of combat, that is often but not always musical in nature. Plenty of musical alarums exist in Shakespeare's stage directions, but plenty of not-necessarily-musical examples exist as well. In their exhaustive Music in Shakespeare: A Dictionary, Christopher R. Wilson and Michela Calore write, “The sounding of alarums by various instruments, especially trumpets, drums or bells is connected with military atmospheres” (p. 12). In Caesar, the term only appears in act 5, as the two armies battle.
On the other hand, flourish indicates the entrance/departure of the most important characters. In the histories it typically designates royalty, but in Caesar it signals the title character, Brutus, and the soothsayer. Some confusion exists as to flourish instrumentation. Wilson and Calore write oxymoronically, “the flourish was invariably for trumpets or cornetts sometimes accompanied by drums. Some examples of flourishes for recorders and hautboys are also observed. In general, any instrument or instruments could play a flourish as a short (improvised) warm-up to a longer notated piece” (p. 179). Regardless, flourishes were “the most improvisatory and musically least structured” stage direction (p. 179).
Edward Naylor, in his book Shakespeare and Music cites 68 instances of the term “flourish” as a stage direction in 17 of Shakespeare's plays: 22 for the entrance/exit of royalty, 12 for the entrance/exit of important non-royalty, 10 for the public welcoming of royalty or otherwise important characters, 7 to conclude a scene, 6 to indicate victory in battle, 2 to announce royal decrees, 2 to indicate entrance/exit of a governmental body (senate, tribune), and 2 to signal the approach of a play-within-the-play (p. 161). (How come that only totals 63?)
“The most exclusive, least used signal in military and courtly contexts”, a sennet is similar to a flourish, but more standardized (less improvisatory) and longer in duration (Wilson and Calore, p. 376). According to Naylor, “sennet” occurs only 9 times in 8 plays – 3 to designate the entrace/exit of a Parliament, 3 for a royal procession, and 3 to signal the presence of royalty (p. 172).
So there appears to be no set definition for these terms, only general patterns in their usage. It should also be noted that there is much debate whether or not Shakespeare actually wrote his own stage directions, or if they were added subsequently by editors. We will probably never know for sure, one way or the other.
While most musical references in Caesar are stage directions and thus aren't necessarily Shakespeare's words, there are three other musical references scattered throughout the script that (presumably) did come from his pen:
While much music based on Julius Caesar exists, little of it bears any resemblance to Shakespeare's play. Georg Frederic Handel's Giulio Cesare, for instance, is the best-known and most frequently-performed, yet has nothing to do with Shakespeare.
Robert Schumann's 1851 Julius Caesar Overture, Op. 128 bears little resemblance to Shakespeare's play, either, though it was apparently inspired by it. In his essay 'Shakespeare in the Concert Hall', Roger Fiske articulates and addresses a two-fold challenge of programmatic music: On one hand, the composer is limited by the narrative functions s/he is supplementing; on the other, the composer must write compelling music, independent of any narrative connotations. “Only when a composer fails at both levels,” he writes, “as Schumann did in his overture to Julius Caesar, is descriptive music best left on the shelf” (p. 181). Ouch!
Orson Welles famously spearheaded a 1937 production of Caesar with the Mercury Theater, featuring music by Mark Blitzstein. That production was adapted for a 11 September 1938 radio broadcast, retaining Blitzstein's score, which is available on Amazon for just $0.69. (The description falsely cites Bernard Herrmann, but the broadcast correctly credits Blitzstein.)
Miklós Rózsa composed the score for a 1953 film adaptation of Shakespeare, directed by Joseph L. Mankiewicz. Its entire soundtrack can be heard on YouTube.
Giselher Klebe wrote a brief dodecaphonic opera titled Die Ermordung Cäsars [The Murder of Caesar], Op. 32 in 1959, with a libretto by August Wilhelm von Schlegel based on Shakespeare. In his essay 'Shakespeare and Opera', Winton Dean describes this work as “a projection of the moral disintegration of the Roman people, suggested perhaps by the collapse of the Nazi regime in Germany. Everything is concentrated on creating an impression of growing chaos” (p. 155).
Winton Dean, Dorothy Moore, and Phyllis Hartnoll's 'Catalogue of Music Works Based on the Plays and Poetry of Shakespeare' also lists the following, which I have annotated, though I've been unable to find any further information:
Curiously, several songs from the pop world relate to Shakespeare's play. Phish's 1994 'Julius' draws on Shakespeare's account, according to The Phish Companion: A Guide to the Band and Their Music. The American rock band The Ides of March (of 'Vehicle' fame) is a direct reference to Shakespeare, stemming from their bassist, Bob Bergland, who read Julius Caesar in high school. There are also several pop recordings based on the ancient Roman dictator that might or might not relate to Shakespeare's play:
A workshop for developing thoughts on William Shakespeare's writings.