I don’t know how controversial it was, but wasn’t “She Bop” by Cyndi Lauper pretty universally known to be about masturbation? Did anybody object to it back when it came out?
I seem to remember it as the sort of thing that you had to be fairly clever and attentive to detect… I myself never knew this until I read a music trivia book much later after hearing the book.
My most vivid memory of a controversial song was Frankie Goes To Hollywood’s “Relax.” It was banned by Radio 1 in the UK, because Mike Read got his knickers in a twist about the lyric "relax, don’t do it, when you want to sock to it… when you want to come."It seemingly became more popular after the ban… and generated all of those crap “FRANKIE SAY” t-shirts.
Ah, the 80s.
I was in my late 30s when it came out & I only found out about it when reading an article on the Net about the abortion theme of the song “Freshmen”, in which “Brick”
was also mentioned.
I remember Tipper Gore getting her panties in a bunch over Judas Priest’s Eat Me Alive, which wasn’t even released as a single. I also remember reading an article that quoted Rob Halford talking about the song. He said something along the lines of “We were drunk out of our minds when we wrote it. We woke up the next day and read what we’d written, and said, ‘Holy [expletive deleted]!’ We wrote that? We censored it ourselves.” And it still got complaints.
I seem to recall some concern about it, but not that much. There were some comments about it, but no one seemed to be up in arms about it. Certainly nothing like the controversy I recall when Madonna released “Papa, Don’t Preach.”
Many radio stations refused to play Billie Holiday’s recording of Strange Fruit when it first came out.
Ah yes, Relax. Mike Read played the song at peak listening time on his BBC Radio 1 Breakfast Show and then said he was looking at the record sleeve and reading the lyrics, and it was the filthiest thing he’d ever read. (I couldn’t help thinking he must have had a somewhat sheltered life.) He refused the play the song again, the BBC started looking into how it had ever made it onto the air, questions were asked in Parliament (OK, I may be exaggerating), etc. etc.
IIRC the whole thing turned out to be a publicity stunt. None of the song’s words taken individually were offensive and unless you were paying attention you couldn’t really tell what the lyrics were. Read’s action did generate huge publicity for the song, and I think it became a tactic for record companies for some time afterwards to deliberately get songs banned by the BBC thus ensuring that the great British public would rush out and buy them.
Happy days! 
In the UK Paul McCartney/Wings released Give Ireland back to the Irish in 1972.
It stirred up controversy and was banned but was still a minor hit…
Hinkley Had a Vision by the Crucifucks would have offended a lot of people if they had good airplay. Being a moderately successful punk band, not that many people heard the song.
quite a few other punk bands had songs that some people found obnoxious.
Frank Zappa’s “Bobby Brown Goes Down” was, at one point, subject to a blanket ban by Brisbane’s then campus radio station 4ZZZ - a station which usually loved to play “controversial” records.
mm
McDonald’s? Wasn’t that about a school shooting?
I remember when “Eight Miles High” by the Byrds was banned because of its “druggie” theme. Although Roger McGuinn explained it was really about a trip on a Lear Jet.
Was this even released as a single in the US? It definitely didn’t make anywhere near Casey’s Top 40.
Yes, a school shooting, by a young girl- the McDonalds incident (San Ysidro, CA) was way after this song was released.
"They’re Coming To Take Me Away " by Louis XIV, according to Wikipedia.
"Ohio " by Crosby, Stills & Nash
"Physical " by Olivia Newton-John.
San Ysidro McDonald’s massacre, James Huberty thought Wednesday was a pretty good day for “hunting humans”.
CMC fnord!
Grace Jones’ “The Bullshit Song”
Lene Lovich’s “Lucky Number”
Oh, and “Turning Japanese” 
Huh. We used to dancercise to “Physical” during high school P.E. class, and we used to dance to “Turning Japanese” at school dances. Never thought anything of it.
And “Relax”.
And the party lyrics to “Mony, Mony”.