One of the first works you'll see after entering the Barbican's latest exhibition is Richard Hamilton's Just what is it that makes today's homes so different, so appealing?. Be careful not to miss it, because despite being one of the smallest items on show, it is perhaps the most important.

This is a 1992 print of a 1956 work, a montage of a domestic scene, and the image which was at the forefront of a whole new artform. A housewife Hoovers a staircase; a reel-to-reel tape recorder sits proudly on the floor; a blown-up cartoon drawing is hung on the wall; the face of a beautiful actress is portrayed on TV; a posturing naked lady reclines on a sofa; and in the centre of the room stands an all-American beefcake grasping an enormous lolly with "POP" written on it.

This was the image which presented the constituents of pop-art to the world: high/low cultures, technology, mass-media and advertising, cartoons, sex and, above all, humour. That the work is by a British artist using imagery from American magazines also suggests the universality and globalisation of the movement which would follow in its wake, and which still has a legacy in current art practice.

All the subjects listed above and more are picked up in this new exhibition of wide scope, and Hamilton's approach of cramming a scene with visual information has also been followed here – there is lots to look at! The whole downstairs gallery, devoted to Pop Beginnings and Pop Icons, has been turned into a cornucopia of widely sourced works. A whole section, behind a shop front-inspired glass screen, is devoted to Andy Warhol's and Alexander Girard's brand influenced ideas, and alongside these is Britain's own twist – a 1961 work by Hockney, Tea painting in an illusionistic style, a painting of an oversized Ty-Phoo packet with a naked seated man. There are also timeless and instantly recognisable pieces – Gaetano Pesce's scaled-up Anglepoise lamp, Moloch (1970–71); Lichtenstein's In the car (1963); and others such as Yonel Lebovici's Fiche male (1977), a massive plug which is itself producing light – haven't lost their wit in the intervening decades.

The premise of this exhibition, however, isn't just to show seminal pieces but to investigate the links between Pop Art and design of the same period – the post-war years of excitement, opportunity, new ways of thinking, consumerism and seduction. And, seemingly, chairs. This is a touring exhibition initiated by Vitra Design Museum, who hold possibly the largest archive of chairs in the world, and they have drawn heavily from their repository for this show. That the "chair" comes to dominate design discourse is a bit of a disappointment, especially in this instance where I can't help but think it would have been more interesting to see the everyday designs of the period, which came from the same conceptual ingredients as Pop Art. The Braun T3 pocket radio or the BBC test cards would have slotted in perfectly and helped provide a bit more cultural context for the artworks.

Upstairs the exhibition breaks into individual spaces to concentrate on thematic issues, but after experiencing the excitement and busyness of the lower galleries these areas seem very sparse and sad. It does give space to look into specific media and approaches such as photography, including Ed Ruscha's Every building on the Sunset Strip (1966) and images from Robert Venturi's and Denise Scott Brown's Learning from Las Vegas (1968), but tucked away up here these items are difficult to read within the wider context. The thematic approach as opposed to the chronological one works in places, but more so downstairs where your gaze crosses a number of items and starts to find its own relationships between pieces. Upstairs, everything exists in its own distinct world and without much contextual information or cross-referencing, the show really loses energy.

This is nowhere more evident than when addressing architecture. Five small images alongside Smithson's film The house of the future (1956) are squeezed into one of the gallery's awkward corridors – scant space for a significant design discipline. Oh for a mock-up of the BT Tower's spinning restaurant, or for Archigram drawings instead of yet another room devoted to chairs!

Also lacking was much consideration given to how the legacy of Pop in both art and design still resonates. A space containing a few pieces from the transition of Pop Art to postmodernism is the final room, but it left more questions than answers.

This is a very good collection of pieces and you won't leave short on visual stimulation, but with a slightly more open curatorial brief and a less restrictive pool of objects from the 'design' side, it could have been a great exhibition.

Pop Art Design, at Barbican Art GalleryWill Jennings reviews Pop Art Design at the Barbican Art Gallery.3