To the 18th century Englishman, opera was a decadent Italian export, and this fad for opera among the rich was nothing so much as killing fine traditional English theater like Shakespeare. (take a look a what's in the wastepaper wheelbarrow)
To begin with, opera is sung in foreign, specifically Italian, which is the worst sort of foreign, it's Catholic foreign. 18th century England was very anti-Catholic. Homes of Italian opera singers were targeted during the Gordon Riots, though the Wikipedia article doesn't make mention of this. Secondly, Italians and Catholics are associated in the English mind with sodomy, an exotic, Eastern practice spreading ever westward through, yes of course, Catholics.^1 The 1730s, when London would for the first time support two opera houses at once (if not for long), was also the height of the work of the notorious Society for the Reformation of Manners and their raids on mollyhouses (gay brothels.)
Sodomy of course goes hand in hand with another decadent Eastern practice the opera was flaunting about on stage - eunuchs. Eunuchs, with their unspoken promises of risk free, pleasure driven sex, where a threat to the virtue of English women.^2 So threatening that anti-eunuch propaganda was quickly translated from the original French to English so its important message of the ultimate downfall (and sexual unfulfillment) awaiting any woman who dared to fall in love with a castrato could be disseminated.
I'll sum it up with this selection from a poem of 1726:
I hate this Singing in an unknown Tongue
It does our Reaſon and our Senſes wrongs
When Words inſtruct, and Muſic chears the Mind :
Then is the Art of Service to Mankind :
But when a Foreign Ox, of monſtrous Size!
Squeaks out a Treble, Shrill as Infants cries,
I curſe the unintelligible Aſs,
Who may, for ought I know, be ſinging Maſs.
Opera was full of eunuchs, sodomites, and Catholics, and you really couldn't trust it.
For a good overview on the relationship between the English, Italian opera, and sodomy, look at the first chapter of Handel as Orpheus although it is covered in other works too.
For more on the castrati as a sexual idea and that idea's relationship with English femininity, see "'There will be all the world there': Sexual Trouble and the Fans of Farinelli in Henry Fielding’s The Historical Register" by Liberty Smith, published in this book
Plato tells us (in book III of the Republic [I am not knowledgeable in the classics, I just linked that translation for convenience]) of the importance of the arts because of how they affect people. Music (which was probably very related to poetry) was considered particularly powerful because it was considered to be able to greatly affect people's emotions and have a strong influence in behaviour and even character.
Music was to be considered an important part of the education of the citizens. If you exposed people (particularly the young) to the "wrong" kind of music, you could be affecting those people deeply enough to get the "wrong" kind of citizens in the future. At first you were supposed to expose people only to the "good" music, the one that would affect people in a positive way. Virtuous people with strong character could take the evil stuff, but you were not supposed to throw that kind of thing at those who were not ready.
What kind of music was he talking about? It is not so clear. We don't not know that much about ancient Greek music... He makes references to music that used some specific scales (which could have had some specific mannerisms attached), but there's nothing telling us exactly what was good and what was bad.
Getting into religious examples, Maimonides (a Jewish scholar, 1135-1204) provided a systematic summary of prohibitions against music. In increasing order of severity we have: listening to a song with a secular text, listening to a song that is accompanied by an instrument, listening to a song whose content includes obscene language, listening to a string instrument, listening to passages played on such instruments while drinking wine, and listening to the singing and playing of a woman (Music And Manipulation: On the Social Uses and Social Control of Music, PP 243). Other religious folks have had issues with specific kinds of music, too.
You mention jazz, rock and rap as examples in the 20th century. I'd like to mention other 20th century situations.
Joseph Stalin also thought art had a very powerful influence on people, because it was a very powerful medium for propaganda. He went for a bureaucratic control over the arts, and "formalist" music was not allowed. Formalist music would be music that was conceived in terms that exclude the cultural context, individualistic, non-representative (too abstract), difficult to be understood by the proletariat. It was decadent and unintelligible bourgeois art that had nothing to offer to the people. Some of the music of Dmitri Shostakovich, Sergei Prokofiev and Aram Khachaturian was considered formalist. Religious music was also considered bad, and anything that could be considered anti-revolutionary was bad.
What was considered good music? Music close to the ideals of socialist realism. It was supposed to be understandable by the people, and should tell good things about the revolution. Things like rapid progress, the achievements of the workers, the great plans for the future, good things about great figures, etc. The Grove's entry on Socialist Realism provides sources on this.
The Cultural revolution in China was also at some points against the corrupting influence of Western music. They closed music schools and sent people to work camps to be rehabilitated.
As /u/hillsfar mentions in another comment, the Nazis were doing something similar. I guess it's not necessary to mention the work of musicians of Jewish origin was not approved by them.