We lovers of the continuing Design Museum soap opera have to hand it to the scriptwriters. This was a drama that started brightly, back in 1987, then rather lost its way. To be honest, I’d stopped watching. But the past few years have been great. We have seen daft shows, rows, the resignation of the chairman, a bonkers design award and now the sudden departure of the museum’s heavily criticised but always entertaining director. Is it in trouble? Not a bit of it. They are going to build a bigger, better, £50m Design Museum. I like that attitude.

The museum has had several directors, but only two high-profile ones: the first, Stephen Bayley, and the most recent, Alice Rawsthorn. Both fell out with their employers and left somewhat abruptly. Terence Conran, whose Conran Foundation effectively underwrites the museum, is not someone who likes a cult of personality to build up around his directors. His on-off, sometimes vitriolic relationship with Bayley, who had originally been his protégé, is the stuff of legend. Having overseen the building of the museum — a low-budget conversion and extension of a low 1940s warehouse in Bermondsey into a kind of 1930s-modern pastiche — Bayley left in 1989.

Rawsthorn, who arrived in 2001, was given more of a chance than observers expected. Even after the museum’s chairman of trustees, the designer-industrialist James Dyson, resigned in protest over her exhibitions policy in September 2004, Rawsthorn stayed. Somehow she had managed to sideline the trustees, even Conran himself, becoming, in effect, the sole decision-maker. Attempts were made to resolve the crisis. But now she has gone, with she and the museum saying effusively complimentary things about each other in the sort of language that suggests the careful scrutiny of lawyers. So, what’s next?

It does not have to be like this, of course. Rawsthorn’s predecessor, Paul Warwick Thompson, quit surrounded by neither a blaze of fireworks nor a haze of legalspeak. He left simply to get a better job, running the Cooper-Hewitt national design museum in New York. Thompson is credited with rescuing the museum from debt by means of such crowd-pulling shows as The Power of Erotic Design. However, those seeking the thrills and spills of the Design Museum soap opera found little to satisfy themselves in the Thompson era. He seemed to be a safe pair of hands — and where’s the fun in that?

In contrast, Rawsthorn, a noted fashion aficionado, put on a show of Manolo Blahnik shoes. Snipers cattily remarked that most of them could have come from her own collection. Then she famously decided to mount an exhibition devoted to the 1950s flower-arranger Constance Spry, clearing out much of the museum’s historic collection to do so. Was that design? It was the last straw for Dyson, and he left. Then there was the matter of the Designer of the Year award. Fine in principle, but there was an almighty row when, in June last year, it was awarded to a Design Council official, Hilary Cottam, rather than to an actual architect or designer. The scheme she was principally credited for — as a kind of mentor — was the excellent Kingsdale School project, in south London, by the architects de Rijke Marsh Morgan, who were not best pleased. With £25,000 involved, there was more than hurt pride at stake.

All these capers certainly served to bring the Design Museum publicity, which it desperately needed. In particular, the Designer of the Year award, which had been a notably lacklustre thing, got acres of newsprint. Similarly, Rawsthorn got visitor numbers rising — though, as Dyson likes to point out, that was only relative to the slump following 9/11. Nobody could pretend things were hunky-dory.

Yet the good news in all this is the Design Museum’s plans for the future. The existing building, which, all those years ago, in the postmodern 1980s, was something of a breath of fresh air, does not exactly command the public’s affection. It is too small, too lacking in character and slightly out of the way, for all that it is an adjunct to Conran’s successful “gastrodome” Thames-side restaurant complex. As to where its £50m replacement is to be built — by 2012, allegedly — that is not being divulged. Conran has in the past talked about the attraction of including a design showcase in the huge commercial redevelopment in and around the old Battersea Power Station, which is being built on much the same timescale.

It would be good to see Dyson back, if not necessarily as chairman: his successor, the financier Luqman Arnold, is well regarded. “I adore the museum, and was sad to leave it,” says Dyson. “Whether or not I go back is entirely up to the trustees and to Terence.” That sounds like a yes to me.

Whatever form the new Design Museum takes, it still has to confront the perennial problem of design — what exactly is it? Dyson remarks that the museum had swung away from Conran’s original idea for it as a shrine or interpretation centre for manufactured goods. Maybe that was no bad thing. Design was always a broad church, and it is getting broader. It encompasses 3-D product design of the kind that Conran and Dyson are known for, graphics (from newspapers to television idents), computer games, trains, planes and automobiles, the animated rock band Gorillaz, engineering, posh frocks, home makeovers, even cookery — and, at a pinch, flower-arranging.

Everybody understands what a building is, even if they don’t understand why it looks the way it does. Design is a much more diffuse and slippery subject. Unlike architecture, it is a trade rather than a profession, requires no particular qualifications to practise and means just about anything. This is the legacy of the nostalgic Victorian era, when Ruskin and Morris got all misty-eyed about their notion of the honest craftsman best left untrammelled by wing collars and stovepipe hats.

Maybe they were right, but one of the results of that laissez-faire attitude is that nobody can really speak with authority for all of design. When the best interface you have with the public is a motley collection of style commentators and government bureaucrats, then you know there’s a problem. A revitalised Design Museum, with a high-visibility director — ideally, one who avoided becoming a self-parody — would go some way to redress the balance.