Here's a look at some previous censorship scandals:
SEX, DRUGS AND DOUBLE ENTENDRES
The 1960s and `70s saw several songs barred from airplay for sex or drug references, including The Beatles' "A Day in the Life," for a fleeting and implicit reference to smoking marijuana. For The Kinks' 1970 hit "Lola," the trouble was not sex or drugs, but product placement. The line "you drink champagne and it tastes just like Coca-Cola" fell afoul of the public broadcaster's rule banning corporate plugs. The brand name had to be replaced with "cherry cola" before the song could be aired. The BBC frequently has been targeted by self-appointed moral guardians, most famously the late anti-smut activist Mary Whitehouse, who campaigned for decades against what she saw as pornography and permissiveness.
In 1972, Whitehouse got the BBC to ban the video for Alice Cooper's "School's Out" for allegedly being a bad influence on children. The controversy helped the song reach No. 1 in the charts, and Cooper sent Whitehouse flowers. He later said she had given his band "publicity we couldn't buy."
But Whitehouse's campaign to get Chuck Berry's "My Ding-a-Ling" banned on grounds of indecency was unsuccessful. The BBC's chief at the time told Whitehouse that, while the song's title could be seen as a double entendre, "we believe that the innuendo is, at worst, on the level of seaside postcards or music hall humor."
Paul McCartney may now be the cuddly elder statesman of pop, but his first single with the band Wings, "Give Ireland Back to the Irish," caused a storm.
Written after the 1972 killing of 13 Irish nationalist protesters by British troops on "Bloody Sunday" in Londonderry, the single was barred from all TV and radio airplay in Britain - but reached No. 1 in Ireland, where it was not banned.
The Sex Pistols' "God Save the Queen" - with its opening refrain "God save the queen, the fascist regime" - was released in 1977, the year of Queen Elizabeth II's Silver Jubilee.
The BBC banned it on the grounds of "gross bad taste," and some stores refused to stock it, to the delight of the punk band, whose anti-establishment credentials were cemented by the controversy.
It remains one of the most famous songs never to reach No. 1 on the charts. It hit No. 2, but was kept from the top spot by Rod Stewart's "I Don't Want to Talk About It."
Punk fans sensed a conspiracy, and debate still rages over whether the Pistols' song really did reach No. 1.
BUOYED BY A BAN
"God Save the Queen" and "School's Out" aren't the only examples of how an airplay ban can boost a song.
In 1984, BBC DJ Mike Read pulled the plug on "Relax" by Frankie Goes to Hollywood midway through its first broadcast, calling the thumping, lyrically suggestive song obscene.
Though it wasn't officially banned, the BBC did not play it. The controversy helped push the song by a then-unknown band up the charts, where it stayed in the No. 1 spot for five weeks.
CENSORS AND SENSIBILITIES
While the moral panics of past eras can seem ridiculous, this week's Thatcher controversy shows that the central issue - which is worse, censorship or causing offense? - is both complex and unresolved. In 2007, the BBC censored the Christmas song "Fairytale of New York" by The Pogues, which was first released 20 years earlier, by dubbing out an anti-gay slur. Some listeners were outraged, but others, including gay rights campaigner Peter Tatchell, said the BBC had been right to remove the word.
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