Within one minute of being seated at Buck and Breck, a cocktail bar in Berlin, I am asked if I want an ashtray. It's a shocking offer for a resident of a city with a smoking ban. Ashtray? Smoking? I look around. Yes, there are smokers, with personal ashtrays.

Indeed, during a four-day bar crawl through Berlin, I leave most bars smelling of cigarettes, a sensation that both my dry-cleaner and I had almost forgotten.

Smoking is one of several ways that the city's cocktail scene, while thriving and full of choice, retains roots in the past. And it does have a past. Berlin doesn't get as much press as other havens of modern mixology such as New York, San Francisco and London, but several of its most prominent bars are as old or older than their counterparts in those cities.

Buck and Breck, while probably the Berlin cocktail den with the most international name recognition, is actually a bit of a newcomer. Its hard-to-find door opened in 2010, and it is about as Berlin-ish as you could imagine. It's stark, but chic; austere, but casual; serious, but hip. The bar, with its nice clean lines, is poised in the front room in such a way that the bartender can't help but be the centre of attention.

Not that you will have much to do with the person making your drinks. Attentive servers act as your conduit. The bartender – a muscular man in a black T-shirt the night I am there – has no time to talk. He has work to do, pulling from two wells full of bottles with colour-coded necks and eyeballing his pours with great accuracy.

From the menu of classic and original cocktails, I try an Ohio (a kind of rye Manhattan topped with champagne), a Beuser and Angus Special (a sort of chartreuse sour) and a Twin Lions (which contains bourbon and scotch). Each is flawless.

Class cocktail lounge

Gonçalo de Sousa Monteiro, the owner of Buck and Breck, once worked at Victoria Bar, a long space with a high ceiling and a bar so long that you could bowl on it. The classic-cocktail-bar look of Victoria Bar is offset by modern art on the walls. It's an aesthetic juxtaposition common to many of the city's cocktail bars, a reflection of Berlin's mix of history and modernity.

The drinks menu, bound in red leather, contains hundreds of choices, including several that were invented at Victoria Bar. White-jacketed bartenders will guide you through the maze of options. I settle on the Boi Portugues, an intriguing aperitif cocktail made of dry port, Campari, vermouth and bitters.

Older than Buck and Breck is Green Door. The door is green, as advertised, and the colour motif extends inside. As at many Berlin bars, you have to push a buzzer to gain entrance. The room looks like a scene designer's vision of a classic cocktail lounge: long, narrow, slightly art deco, dimly lit and filled with well-behaved people. My chosen drink, the suave Velvet Cordon, made of gin and fir needle cordial, seemed a perfect liquid companion to the surroundings.

To get a fuller sense of Berlin's history, there is Bar Lebensstern, on the second floor of a 19th-century villa that escaped Allied bombing during World War II. The house was owned by a Jewish couple but was seized by Joseph Goebbels, who installed in it an actress who was his mistress. The beautiful interior is made up of many rooms, each with red walls and filled with cabinets holding rare, expensive spirits. The menu features a Ranglum, a modern rum drink created by de Sousa Monteiro that is a local favourite. The bartender tells me that most "better cocktail bars" know how to make it.

Stagger Lee, in contrast, doesn't give a hoot about Berliner history. It's done up like the well-upholstered parlour in a high-quality bordello in an American frontier town. Swinging doors lead to the restrooms. The menu focuses on classics, but with a house twist: Lee's Boulevardier, Stagger's Mai Tai and so on.

Thank you for smoking

The ghostly image of Samuel Beckett's face in a window is the only indication of Becketts Kopf's location. The darkened bar is filled with deep armchairs ripe for deep discussions, and the menus are tucked inside out-of-print copies of a German translation of a book on Beckett written by the American theatre critic Mel Gussow.

If the place has a sense of humour about itself, it's so dry as to be imperceptible. When I enter, I am asked if I have a reservation. I look around at the near-deserted bar. Are all the seats reserved? "No," the young bartender says. "But there is a reservation at 11." It's 7pm. I sit down.

All chilliness is forgiven, however, upon first sip. The laconic bartender bends over his work, and the care he takes shows in the excellent results. All are accompanied by an exceedingly small glass of water, another peculiar earmark of Berlin cocktail bars.

The ultimate exemplar of Berliner bar quirkiness, however, has to be Rum Trader. The bunker-like shopfront, on a quiet corner in a residential neighbourhood, offers no indication of the unique lunacy within. About the size of a key-making shop, it has maybe 12 seats. The single table is occupied by four cigar-puffing men. Soft jazz plays. The bartender inquires if I want a rum drink or a gin drink. When I say rum, I get a Hemingway daiquiri. When I say gin, I get something with gin, lemon juice and crème de cassis.

At the bar, a garrulous man nurses a blood and sand. "The are no cellphones here," he says. "You're not on the clock. You're here for talk." And talk (and talk) he does.

Within its four walls, Rum Trader feels completely cut off from the rest of Berlin, or the world for that matter. It is a love-it-or-leave-it kind of place. I kind of love it for its insistently odd attitude. But after two drinks, I am ready to leave it. I keep the smoke for a souvenir.