Mad Men has a lot to answer for. The cult television sit-drama has for the past few years been catapulting us back to the Madison Avenue of the late Fifties and early Sixties and connecting us to social, sartorial and behavioural traits that seem rather appealing in these austere times. Chain-smoking from the moment you wake up is one of them. Drinking extravagant cocktails through the working day is another, starting at around 11am.

It is no coincidence, therefore, that cocktail bars on both sides of the Atlantic are reporting a considerable surge in interest. According to some of the celebrity bartenders and mixologists emerging in this booze boom, we are in the throes of the third golden age of cocktails. The first was in the 1920s and 1930s, during Prohibition; the second was in the Mad Men Manhattan of the late 1950s; the third is now, in the recession of the early 21st century, when we find ourselves back in the cocktail bars, sipping on dramatic social fuels and sharing our collective pain in dark rooms where pianists are vamping standards from the jazz age.

So it was that, some time last autumn, I packed all this cultural baggage into a small case and hopped this way and that across the Atlantic, investigating martini bars and barmen. While there are great examples in Paris (the Bar Hemingway at the Ritz), Milan (the Principe Bar at the Hotel Principe di Savoia), Vienna (the Blue Bar at Hotel Sacher), Hong Kong (The Mandarin) and, of course, Chicago (the Coq d'Or at The Drake hotel), the two great centres of the genre are New York and London.

My original idea was to find the perfect martini, the drink that in all its purity and simplicity seems to epitomise the cool cocktail life – and indeed, I kept that in mind throughout my travels. Some say the martini was named after the Martini-Henry rifle because it had a similar kick, others that it was in tribute to Martini di Arma di Taggia, the barman at the Knickerbocker Hotel in New York who first put together dry gin and dry vermouth. The drink's celebrity was secured, of course, by James Bond coolly asserting that a martini should be shaken not stirred – an instruction that now sends shivers down the spine of any self-respecting barman.

However, as my journey of discovery unfolded, it became increasingly clear that the venue was almost as important as the cocktail itself. Then, inevitably, other concoctions came into the picture, such as the Ning Sling, the Winston and the Crumble. As my knowledge grew, this ended up an altogether more rounded investigation.

I started out in New York, at an old favourite of mine – Bemelmans Bar at The Carlyle on Manhattan's Upper East Side. This really is the archetypal New York cocktail lounge, with a bar against the far wall and a grand piano in the middle of the room, surrounded by clusters of low tables around which huddle regulars and tourists, Upper East Side old money and bridge-and-tunnel new money. It is dark and intimate, with a constant supply of martinis.

The place is named after Ludwig Bemelmans, author and illustrator of the Madeline children's books, a long-time resident of The Carlyle who agreed to illustrate the walls of the bar in exchange for room nights. His drawings provide a surreal counterpoint to the old-fashioned, rather regal decadence of the place.

It was here that I met up with Brian Van Flandern, one of the city's most celebrated mixologists and the cocktail consultant at Bemelmans. He arrived with a hand-tooled leather case, much like that of a doctor on a house call. Before beginning his demonstration, he ceremonially opened the case to reveal a collection of stirrers, strainers, spoons, mixing receptacles and everything else you need to make the perfect martini, all meticulously arranged inside. He then proceeded to mix and lecture me in equal, but perfectly measured, proportions.

"Despite what James Bond said, a great gin or vodka martini should be stirred not shaken," Van Flandern insisted, as he began to prepare the perfect Bemelmans martini. "The best gins and vodkas have an oily, creamy quality to them – so when we shake them, we are forcing micro-bubbles through the liquor and changing the viscosity, the texture of our cocktail. We want that oily coating to cover our palate so that, when we exhale, we can fully enjoy the spirit."

Van Flandern mixed my Ketel One vodka martini in a glass shaker using crystal-clear ice, and stirred it for half a minute. He then poured the mixture into a pre-chilled glass with a swirl of vermouth added to coat its inner surface. The final ingredient was a lemon twist which, according to Van Flandern, "caramelised the oils".

As I sipped, the pianist played "Take the 'A' Train" in the background, the Bemelmans crowd chattered away, and the martini master continued with his soliloquy. "Most people prefer vodka martinis to gin because they find gin too aggressive on the palate," he told me. "But when a gin martini is properly made, properly stirred, diluted and garnished, gin is a lovely experience. You get all the rich botanicals without the bite."

I could have stayed at Bemelmans all night but I was on a mission. Instead of taking up Van Flandern's offer of another martini, I headed south to Salon de Ning at The Peninsula hotel. While Bemelmans is a traditional dark, intimate room with piano music, Salon de Ning is contemporary, funky and largely outdoors. On the 22nd-floor deck of The Peninsula, you find yourself sitting in the middle of a forest of Manhattan skyscrapers. Here there was no pianist, just cool music piped through a speaker system. Nor was there the messianic attention to detail that I'd found at Bemelmans, though the barman, Ron Pellegrino, offered up some pretty convincing cocktails.

A traditionalist, he said he preferred straight-up gin martinis – Farmer's, Hendrick's and Beefeater were his top choices – but his clientele seemed to choose fruitier concoctions. The signature drink here is the Ning Sling (Absolut Mandarin vodka, Soho lychee liqueur and fresh mint leaves shaken with lychees and passion-fruit juice), while Passion Martinis, Blueberry Slings and Gingerpolitans are all multi-coloured alternatives in which lashings of fruit juice disguise the taste of the alcohol.

I had Pellegrino mix me a Farmer's gin martini and, fresh from the Van Flandern lecture at Bemelmans, I was taken aback as he shook the gin in a stainless-steel mixer with vigour and commitment. "We shake our martinis to get them super-cold," he said without pausing, then poured the spirit into an ice-cold glass, adding a sprig of lemongrass. It was absolutely fine, had the desired effect of making me feel at peace with the world, and made me wonder whether all this talk about viscosity and perfect aeration is a lot of old mixology mythology.

Now it was time to cross the Atlantic. Van Flandern had cited the American Bar at the Savoy (although he had not been there since it had reopened) and The Connaught as two places of excellence in London, while others claimed they'd had their best martini at Dukes Hotel.

I started at the American Bar. Like Bemelmans, this is a place with considerable history. It opened for business in the 1890s but became famous at the end of the Roaring Twenties when Harry Craddock fled Prohibition in his native America and set up his practice at the Savoy, thereby dramatically shaking and stirring the London cocktail scene. Craddock invented the White Lady (dry gin, Cointreau, lemon juice) and brought with him the dry martini. He set out his recipes and tips in The Savoy Cocktail Book, first published in 1930, which remains, after many reprints, the timeless bible of the connoisseur.

The American Bar has, like the rest of the Savoy, been subjected to a makeover. When it re-opened last October people feared the precious ambiance and art-deco feel would have been diminished. Fear not, for it has retained much of its glamour: the bar still glitters with art-deco flourishes, the walls remain adorned with Terry O'Neill's black-and-white photographs of the Hollywood stars who once graced this bar, and the white piano is still the focal point. The only problem with the new American Bar is that there is a queue to get in every night, detracting from any impression of sophistication.

At the bar I met up with Erik Lorincz – an award-winning barman, formerly of the Connaught Bar – who, when asked which gin makes the perfect martini, responded without missing a beat: "Some women like wearing Christian Louboutin, others Manolo Blahnik, some men choose a BMW, others a Mercedes-Benz."

I chose Tanqueray No 10, although it could have been Beefeater 24, Hendrick's or any of the new American micro-distillery gins that are fast becoming fashionable on both sides of the Atlantic. How much vermouth should one add? Again the smart Mr Lorincz had a snap answer: "It is a question of how dry you want your martini. If you like it wet, we keep the vermouth in. If you like it dry, we rinse the glass with vermouth. If you like it extremely dry, I call my friends in Singapore and they put the vermouth next to the phone while I'm stirring the martini."

With an extravagant gesture redolent of Tom Cruise in the film Cocktail, Lorincz transferred the delicately prepared liquid into the chilled martini glass and I took my first sip and confirmed the opinion shared by all martini men – while there is nothing wrong with a vodka martini, the more traditional gin version really is the great example of the drink.

The following weeks were spent further pursuing the perfect cocktail. At Dukes Bar, I met up with the famous Alessandro Palazzi, who offered to mix me a truffle martini – an offer I declined, having had the infusion passed under my nose. Instead I went for Palazzi's second option, which was a Tanqueray 10 Gin Martini, poured into a refrigerated glass served not at the bar but at my table. Dukes Bar is an intimate venue – some would say crowded – and, according to Palazzi, many of its customers turn up in black tie to sip their martinis. He said he and his staff serve between 150 and 200 martinis a night, proof indeed that the third golden age of the cocktail bar is upon us.

The Connaught Bar is probably the most celebrated in London – and rightly so, given its classy decor, the generous layout of the room and the dominant presence of head barman and mixologist Agostino Perrone. It really is the most seductive place in London to sip a cool martini. (The only thing I didn't like was the piped music – get a pianist.)

When the bar re-opened after a serious makeover in 2008, it was relaunched by Perrone and his cocktail twin Erik Lorincz (see above) and they lured the cocktail cognoscenti back with a brilliant array of colourful beverages and, most importantly, the addition of a martini trolley. The trolley itself – black lacquer and leather – is a stylish statement, but the ingredients contained therein are what really count. Perrone's specially-mixed vermouth and his range of infusions (cardamom, coriander, lavender, ginger, vanilla, and so on) are what make this bar's martinis so special. And don't bother trying to replicate this at home – I did, and failed dismally. My advice is to go to the Connaught Bar and have the white-gloved mixologists do it for you. It makes for a memorable experience.

Finally, at the last stop on this epic journey of discovery, I found myself at the Ai Fiori bar in midtown Manhattan, where I met up with a man who does to alcoholic beverages what Heston Blumenthal does to food. Eben Freeman, who dislikes the term mixologist, is a cocktail creator with outlandish ideas, such as putting bacon into bourbon in a process he calls "fat washing" – apparently because "the molecular structure of alcohol takes flavours from fats very well". A few years back,he created what he called the "mojito of the future" using a liquid nitrogen technique called spherulation.

In yet another variation of the martini-shaken-or-stirred conundrum, Freeman has taught his bar staff to shakethe vodka martinis and stir the gin martinis. By the time he came around to mixing me a cocktail, I was both bored with the debate and actually desperate to drink something other than a martini. So it was that he made me a Crumble, using fat-washed rum, brown butter and falernum (a cane-based liqueur from Barbados), poured over a huge cube of ice and topped with apple cider from Normandy and a sprinkle of nutmeg. It was quite superb – and I have to say, it left the martinis I'd been tasting over the past few months way behind.