There’s no such thing as bad publicity, they say. But I wonder whether Zaha Hadid – of whose past, present and future work a near-comprehensive survey opens at the Design Museum on Friday – would have agreed ten years ago, when she was on the wrong end of one of the most humiliating public snubs in the history of modern architecture.

She had beaten 250 other architects to win the competition to design Cardiff’s Opera House. Her entry, more like a giant glass necklace than any known theatre, caused a sensation. Unfortunately, the sensation was almost entirely negative, at least within Wales. The local press vilified the design and its maker. Populists claimed (wrongly) that if Cardiff built this “elitist” opera house it wouldn’t get its coveted new rugby stadium. Opera singers complained nonsensically that people would be able to see them undressing. And the Millennium Commission, which was stumping up most of the money, applied the coup de grace by describing the design as “insufficiently distinctive”, which was a bit like saying that the Atlantic is insufficiently wet.

And who was this Hadid anyway? She was swiftly belittled as a “paper architect”, because only one of her fantastical designs had ever been built – a fire station somewhere in Germany. And she was an outsider twice over: first as an Iraqi, albeit one who had studied and worked in London since the early 1970s; and secondly as a woman crashing into the men’s club at the top of British architecture. And a pretty mouthy woman, too, with opinions as strident as her architecture. She rubbed people up the wrong way.

Well, in Wales in 1996 she experienced a stinging backlash. For about three years afterwards, she later admitted, the stigma was so enormous that it was virtually impossible for her to get any projects off the ground. Indeed, her most prominent completed work in that period was the Mind Zone at the Millennium Dome. That’s how bad it got.

A drawing of the unbuilt Cardiff Bay Opera House is displayed, unapologetically and unashamedly, in the Design Museum show (the first major exhibition under the museum’s new boss, Deyan Sudjic). Indeed, much of the ground floor is taken up with designs for fantastical projects that never left the drawing-board.

By comparison with architects such as Lord Foster, who would be unlikely even to embark on a project without a firm commitment to build, Hadid entered the 21st century with just one finished permanent building to her name (the fire station), after more than 20 years in the business.

The most significant of the “paper” designs, perhaps, is a fine painting (Hadid is an international-class artist as well as architect) of the Peak, a gravity-defying apartment complex that she conceived back in the early 1980s for the heights overlooking Hong Kong. A rude riposte to the anodyne PostModernism that was then stifling architectural innovation, it looks like a series of jagged splinters stuck into the living rock. And, as with so much that comes from Hadid’s imagination, it seems partly to belong in some (extremely classy) sci-fi comic, and partly to be the result of some huge geological upheaval.

But if the show’s first part is largely a chronicle of unrealised visions, the rest demonstrates how impressively Hadid, now 56, has made up for lost time in the past decade. She employs 170 people at her Clerkenwell office (a converted school), and you can see why. Yes, her unbuilt projects still vastly outnumber the 15 or so Hadid buildings that actually exist. (Three of the biggest finished works are showcased here: a modern-art museum in Cincinatti; an elevated science centre in Germany that looks like a walking spaceship; and the BMW factory, in which the production-line snakes in and out of the office space so that even the accountants can gawp at the gleaming metal as it passes.)

All that is set to change in the next few years, as Hadid sends her lieutenants scampering with weird and wacky ideas to all corners of the globe. In Azerbaijan there will be a skyscraper for the state oil company; in Kazakhstan a new city square, perhaps the world’s first piece of largescale town-planning entirely devoid of right-angles or straight lines (another Hadid “signature”); in Dubai a performing arts space suspended over the sea; and in Guangzhou an opera house imagined as two giant boulders thrust out of the adjacent river.

These ongoing and future projects, more than 25 of them, are all portrayed on the exhibition’s second floor, sometimes in dazzling, aluminium-based illustrations that seem to glow and vibrate as you pass them. Trust Hadid, the world’s most flamboyant architect, to turn even her two-dimensional drawings into theatrical coups. Architecture exhibitions can be as soulless as kitchen showrooms, but here you can really feel the exuberant imagination behind the buildings.

And the protean energy, too, for the show contains not just architectural models and drawings but also some of Hadid’s startling furniture and knickknacks. There’s even a prototype car, and of course the obligatory handbag. The scale may be miniature compared with her buildings, but there’s no diminution of creative quirkiness, nor of the sense that fluid has been frozen in space. Here are tables, chairs and shelves that seem cast in a single, sinuously twisting curve – like a wave orthe sand dunes that the infant Hadid would have seen in the Iraqi deserts – and then realised in exquisite colours and materials.

Mind you, the prices are exquisite too. Hadid tables tend to start at £100,000. For a mere £130, however, you can now buy a set of her intriguingly off-kilter cutlery (though you get only five utensils for your money).

Perhaps the show’s most intriguing aspect is the evidence it provides that this architectural prophet is at last being honoured in her adopted land. Until last year, when she became the latest big-name architect to design one of Scotland’s Maggie’s Centres (drop-in clinics for cancer patients), Hadid had never had anything built in Britain. Now, however, the reactionaries had better brace themselves: five Hadids are on the way in London alone, not counting her temporary installation for the Serpentine Gallery this summer – a typically ingenious interweaving of giant parasols. They include her eye-popping Olympic Aquatic Centre, as well as a city academy (every fancy architectural practice has to do one, it seems).

Her critics used to declare her ideas “unbuildable”. Now the accusation has changed: her specified materials and designs are said to be so off-the-wall and untested that her projects inevitably run wildly over budget and time. The 2012 Olympics Aquatic Centre, for instance (though scaled down by Government decree), looks as if it will cost double its original £75 million budget. It’s a spectacular concept, like some graceful paper aeroplane – but £150 million for a swimming-pool? You have to ask how.

Another Hadid design, for Glasgow’s new Riverside Museum – set at the intersection of the Clyde and the Kelvin, and aptly shaped like a metallic wave – looks as if it will be three years late and £17 million over budget.

But innovation and ambition never come cheap in architecture. Just read what Wren’s exasperated contemporaries thought about him. Hadid is not merely designing buildings, she is reimagining domestic, corporate and public space. In her breathtakingly sensuous designs such mundanely separated entities as walls and ceilings, indoors and outdoors, cease to be. All are swept into the mighty curve, which itself seems to grow organically out of the landscape. Spine-tingling or spine-chilling, the show is nothing if not a brazen vision of the future.

— Zaha Hadid Architecture and Design, Design Museum, London SE1 (0870 8339955), from Fri to Nov 25. www.designmuseum.org

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