Sixty years ago, in the summer of 1950, a small Californian business was preparing to introduce the world to a new musical invention. The Fender Electrical Instrument Company was based in Santa Ana, 30 miles south of Los Angeles, and it had already come up with the Esquire, an electric guitar that broke with convention by being built from a solid piece of wood. Now, 41-year-old Leo Fender had radically improved on the original to produce the Broadcaster – which, after a spurt of legal hoo-hah, was renamed the Telecaster, and sold to the world.

You know one when you see it: gloriously simple, gracefully contoured, and a byword for how enduring the electric guitar has proved to be. As a sumptuous new coffee-table book titled Fender: The Golden Age 1946-70 puts it: "It is a simple, no-frills instrument, yet still regarded as one of the finest electric guitars ever produced . . . There are very few mass-produced items that can boast the same uninterrupted lifespan." In other words, it beggars belief how an object designed six decades ago doesn't look – or, more importantly, sound – kitsch or outdated. The Telecaster's younger and less elegant sibling, the Stratocaster, tends to go wildly in and out of style, but this guitar remains as unimpeachably cool as ever.

It has long been responsible for the metallic twang that runs across American country, blues, and rock'n'roll. Over here, in the hands of an Essex native named Wilko Johnson, it contributed the distorted buzz to the best records by the British R&B band Dr Feelgood – a sound that bled into punk – and became a constant in 21st-century indie-rock.

The Telecaster unites Status Quo and Hot Chip. It has been the signature guitar of Keith Richards, Chrissie Hynde, Radiohead's Jonny Greenwood and the Clash's Joe Strummer. It's all over records by Elvis Presley, Booker T and the MGs, PJ Harvey, Blur, the Eagles, Manic Street Preachers, and hundreds more. Mention it to its devotees, and they talk about it with an amazing passion.

Sharleen Spiteri, Texas

I'd always played an acoustic guitar as a little girl – there was never an electric guitar in our house. But when I first joined Texas, when I was just about to turn 18, I got my first electric. I knew exactly what I wanted: a black-and-white Telecaster, the same as Joe Strummer. For me, the Clash have always been the ultimate rock'n'roll band.

It was made in 1967, the same year I was born. I've got two spares, but I will do anything not to change my guitar. It goes on tour with me, and it comes home. Even now, I'm getting goose bumps talking about this because I'm thinking, "Some bastard might break into my house and steal my Telecaster."

It's super-light – just the perfect weight. And that nasally, stringy, cutting sound [sings guitar part from Texas's Halo] – it just breaks through everything. But it's weird, because it's not an irritating, high-pitched thing: it always has a warmth to it. It's such a beautiful sound.

I recently did a charity thing with Mick Jones from the Clash. He picked my guitar up, and I was like, "Get your hands off! I don't give a fuck if you're Mick Jones!"

Graham Coxon, Blur

I think my first encounter with a Telecaster was when I had a go on [Blur producer] Stephen Street's, years ago. It looked like a piece of pine with a pretty scratchplate, but it made a really nice sound. I suppose it was the guitar I'd been searching for; I used to draw Teles a lot when I was at school.

It's versatile, simple and strong. You can make it sound old-fashioned and warm, like something you'd have in a doo-wop band, or totally the other way: trebly and trashy. And it's quite difficult to describe, but they have a kind of creak underneath the sound – something you only really get with a Telecaster.

The first one I got was very shiny, butterscotch job: a reissue of one from 1959. I used that throughout the whole of Blur's career. It ended up with a Mr Smiley sticker, and an Air India sticker on it, and a really bad drawing that I did on the back. That was my workhorse, and I've still got it.

It just feels really, really nice: like a BMX, as opposed to a big, heavy bike. The one I use now had been butchered by its previous owner, but the neck was so beautiful, I couldn't resist it. It looked like it had been creosoted, so I call it the Shed.

Francis Rossi, Status Quo

I've got loads of them: three in the room with me at the moment. I got my first one in 1968, in Glasgow. Another guitar broke, and I got it second-hand for either 45 or 75 quid – I can't remember now. I wasn't a good enough player to say, "Oh, I really like this." I just picked it, and got used to it.

Originally, it was sunburst. And then I started to mess with it. I wanted it to be blond, and then natural wood, and then I painted it black, and then green, finally, with a tin of Ronseal paint. I did it on the kitchen table; it's still got marks on it from where I didn't sand it down properly. But that's the one I still play onstage.

When Rick Parfitt first joined the band, he had a Gibson. And then he too bought a Telecaster, and that's the way it stayed. The Telecaster is a byword for Status Quo. They're beautifully simple guitars: Rick's always like, "Just turn it on, thank you." He's even had his tone control deactivated: it's just set to full-on, all the time.

I really love the shape. And it's workmanlike. Solid. Basic-looking. No-nonsense. There's no fragility about it. It's like a tool.

Andy Summers, the Police

I think it's one of the greatest designs of the 20th century. It has never been bettered – partly because it fits the human body so well. With something like a Gibson Les Paul, it's a very heavy chunk of wood. I always had a problem holding it, especially standing up. But the weight of the Telecaster is just right. And its sound just cuts through everything. Everything I did in the Police was on a Telecaster.

I actually ended up with a kind of hybrid model, with some added features. Before I joined the Police, I got it in LA: I was teaching guitar, and some kid came in wanting to sell it. It was pretty beaten up, and I got it off him for 200 bucks. I said, "This is a pretty great guitar – are you sure you want to sell it?" But that was the guitar that transformed my life: it brought me back into playing rock, I came back to England with it, and you know the rest of the story. I would never, ever get rid of it: it's like a great talisman.

In 2007, Fender made 250 clones of my guitar. They took the original to pieces, and photographed it, and made this map of where all the scratches and cracks are. It's like my guitar had babies. I've got six of them. They cost $15,000 each [laughs].