The perfect Private Eye cover makes you gasp and guffaw at the same time. In March this year, after stories appeared in the press about the Duchess of York's financial troubles and Prince Andrew's friendship with a convicted paedophile called Jeffrey Epstein, the magazine displayed the Yorks on thecover in their party finery. A speech bubble had Andrew saying, "Epstein is prepared to give you £15,000," and Fergie replying, "Is that for Eugenie, or for both of them?".

Jonathan Swift, in full "A Modest Proposal" mode, would have approved of that joke. It's shocking, disgracefully unfair, clearly actionable and it skewers the chronically troubled Yorks with admirable economy.

Over the past five decades, royals, politicians, judges, presidents, religious leaders, Eastern potentates, Europhiles, criminals, left-wingers, right-wingers and media potentates have warily approached the news-stands every fortnight, praying that their transgressions and hypocrisies do not feature in the satirical scandal-sheet, nor (please God) their smiling countenances appear on the cover, nailed beyond repair by a killer caption in a rectangular box.

Private Eye is 50 years old this October, and among the celebrations attending its anniversary will be an exhibition at the Victoria & Albert Museum in London. Expect to see plenty of the magazine's distinctive covers here: they represent a half-century chronicle of political scandal and public folly – but also 50 years of exposure, opposition, subversionand ridicule.

The earliest Private Eye covers didn't feature photographs, let alone captions. Issue No 1, published on 25 October 1961, was printed on mustard-coloured paper (a link to the Yellow Book, the bible of 1890s decadence?) and featured just one satirical squib, at Conservative politician Iain MacLeod. A display box announced, "You've been sold a dummy – of what we hope, after further experiment, will be a weekly newspaper to appear regularly in the New Year".

Issue No 3 (30 November) featured a young woman on the cover dressed as Santa Claus, saucily revealing a stockinged leg. A speech bubble issued from her mouth, saying "Ad Maiorem Dei Gloriam". The words ("To the greater glory of God") are well-known to Catholic schoolboys, but seem meaningless in this context. Nonetheless, this was the magazine's Eureka moment: the first photo-with-words joke. From now on, every fortnight for 1,200 issues, almost every cover would carry a topical photo subverted by a satirical headline or a caustic speech bubble.

The next issue carried a photo of the Albert Memorial, headlined "Britain's First Man Into Space"; the appended caption – "He he very satirical" – should have issued from the mouth of Prince Albert's statue, but was given to a random spectator.

Issue No 6 was the first that properly married picture and caption. It showed Michael Ramsey, Archbishop of Canterbury, tête-à-tête with a fellow clergyman, asking him: "And what are you giving up this Lent, eh?" – more nudge-nudge than funny. It's touching, half a century on, to watch the magazine gradually learning to be both sophisticated and devastatingly simple.

Among these early covers, one finds several mis-hits, poor jokes and topical allusions that would mean nothing to later readers. On 5 April 1963, a balding man sitting on a bed says to another man, "And if Private Eye prints a picture of me on a bed, I'LL SUE THEM!". It takes some cudgelling of the memory to recall that the man is John Profumo, embroiled up to his neck in the Christine Keeler scandal. But even when you're not sure of the context, you can feel the teeth of moral accusation.

In a modern world in which any red-top newspaper can abuse members of the Royal Family, and link their names with scandal, sex and drugs, without the least outcry, it's amazing to recall the shock value of the Eye's royal covers. I first saw the magazine in a Battersea newsagents in October 1964 – I was just 11. The cover showed the Queen making a speech for the opening of Parliament and saying, "And I hope you realise I didn't write this crap". A four-letter word! Printed on actual newsprint! Attributed to the Queen! You could feel the ground shifting slightly beneath your feet. You could almost feel centuries of deference starting to give way to something new.

When the magazine put a grinning, toothy picture of Princess Margaret and Lord Snowdon on the cover of issue 226, smiling through an even more shocking caption, you could hear the nation's sharp intake of breath. (Could it be true that the royal marriage was on the rocks? Could Private Eye have broken the news any more bluntly?)

By the mid-Sixties, satire had gone beyond the jokey lampooning of domestic figures into savage international territory. The scribbly, herbivorous cartoons of Willie Rushton were often superseded by the grotesque anti-American rudeness of Gerald Scarfe, whose cartoons, like the Eye itself, displayed an obsession with arses.

A lurking homophobia, characterised by the words "poove" and "ducky" (as seen on the cover featuring David Bailey's famous photo of the Kray twins) was a constant theme. A hot diplomatic embrace between General Nasser and King Hussein of Jordan had Hussein saying, "I'll be buggered if I do this again" – probably the first time the word was seen in print. When Hendrik Verwoerd, the architect of apartheid, was assassinated in 1966, the Eye cover carried the caption "Verwoerd – A Nation Mourns" beneath a picture of four cavorting South African tribesmen. It was a joke that would be applied to many more unlamented dead world figures.

The early Eye years are a fascinating record of British satire taking hold of a new world of posturing politicians, assorted poseurs in the arts world and ill-advised grinning public figures, and roasting them alive with just a few words. Subsequent years saw the brand become more streamlined and polished, but without losing its edge. When the triumphalism of Mrs Thatcher was at its height, during her Falklands adventure, any number of photographs could have been employed to ridicule the gap between her achievement and her striking of heroic attitudes. Typically, Private Eye used the pointing figure of her husband to suggest the couple were on a ferry crossing to the Isle of Wight.

And does any picture encapsulate the dawn of New Labour better than that of Cherie Blair embracing her husband on the steps of Number 10 on 2 May, 1997? "Happy?" she's asking him. "Sorry," comes the reply, "You'll have to ask Peter Mandelson."

Like Wilson, Brown and Macmillan in the 1960s, Heath and Callaghan in the 1970s, Mrs Thatcher in the 1980s, Major, Blair and Brown thereafter, like the Royal Family from 1961 to 2011, like every rogue, pseud and chancer from Robert Maxwell to Rupert Murdoch, you can imagine every object of Private Eye's stiletto-like scorn reading the cover – and wincing with recognition.