One Saturday afternoon the summer I was 16, a couple of friends and I were sitting on a bench in our hometown of Kilkenny, smoking cigarettes and enduring the agreeable boredom of one another's company, when the actor Bill Murray materialised on the other side of the street, wearing shorts and a T-shirt and a baseball cap, and clutching an off-white drawstring laundry bag such as you would ordinarily see in the hands of people who were not major Hollywood stars.

This hadn't happened completely out of the blue – Murray, we were acutely aware, was in town for a new comedy festival – but it was still quite a thing to look up and see him just sauntering down the main street of your hometown like it was no big deal. And so I found myself doing something I'd never have done in non-Murray-based circumstances: I shouted across the street.

"Bill! … Bill!"

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He stopped walking and looked over at me and my two friends. "Hey fellas," he shouted back, in what was unmistakably the voice of Bill Murray, with its low-hanging Chicago vowels and its heavy but somehow inclusive irony. Unable to think of anything else to say, I asked him what he was up to. He hoisted up the bag he was carrying and said, "Just doing a little laundry is all!". Then he saluted, before continuing on his way down the street toward the laundrette, never again to be glimpsed by me or my companions.

By most reasonable measures, this is about as boring a celebrity encounter as it's possible to imagine. The only plausible element of intrigue or drama to be found here is in the unresolved question of why one of the world's most famous film stars would be taking his own clothes to the laundrette, rather than delegating the whole business to the staff of whatever hotel he was staying in. Yet whenever I mention it to people, it tends to get a disproportionately delighted reaction. And this, of course, is because people – people of my generation, at least – tend to relish any opportunity to hear about the various public manifestations of Bill Murray.

This experience of mine counts, in fact, as a very minor entry into the canon of The Bill Murray Story, which is a highly specific and thriving sub-genre of contemporary celebrity folklore. The Bill Murray Story is predicated upon some or other brief but remarkable intervention of Murray into the lives of complete strangers. The setting is usually a public place, and there is often, as we say in Ireland, "drink involved".

The original BMS, the founding narrative of the form, goes roughly as follows: some nameless patsy (a friend of a friend, a colleague's cousin, whatever) is alone on a quiet city street late at night, and is suddenly set upon from behind by a stranger, who places his hands over the person's eyes, tipsy-uncle style. Our flustered protagonist then turns around to find that his or her assailant is none other than Bill Murray; at this point, the star of Lost in Translation, Ghostbusters, and Groundhog Day whispers the words, "No one will ever believe you", turns smartly on his heel and dissolves once more into the unknowable murk of the night.

This story is pretty much your typical urban legend; it crops up in various forms in various places, and its provenance seems to be unknown. Murray was asked about it in an interview with GQ a few years back and, though he didn't outright deny it, his noncommittal answer suggested an unwillingness to impair the progress of a fun story that had taken on a cultural life of its own. "There's probably a really appropriate thing to say," as he put it. "Something exactly and just perfectly right. But by God, it sounds crazy, doesn't it? Just so crazy and unlikely and unusual?"

And maybe what's important isn't the truth or falsehood of the story, but the extent to which people seem to want to believe it, or at least to repeat it. Because even if it's not true to the letter, it's true to the spirit of Murray, who has come to seem less like a movie star than an intermittent event, an ongoing work of living improv. The Murray of popular imagination uses his celebrity as a secular superpower, making unexpected and (broadly) benevolent interventions into the lives of the citizenry. He has made party crashing into a kind of performance art.

A by no means exhaustive list of extracurricular public activities in which, due to photographic or filmic proof, we know him to have engaged over recent years:

*Came across a group of twentysomethings playing kickball at a park on New York's Roosevelt Island, and requested that he be allowed to join in the game.

*Got involved in a karaoke and Chartreuse session with another group of young strangers at a karaoke bar near Union Square.

*Showed up at a student house party in St Andrews, where he'd been playing in a celebrity pro-am golf tournament, had a couple of beers, and then just upped and did the dishes before leaving.

*Strolled into a badly overcrowded and understaffed bar in Austin during the SXSW festival – accompanied, mind you, by his good friends RZA and GZA of the Wu-Tang Clan – and hopped behind the bar and started bar-tending, unilaterally imposing a strict tequila-only service policy.

For a generation of people who grew up watching films like Ghostbusters and What About Bob?, the benevolent intervention of Bill Murray has become a sort of Platonic ideal of the fun experience. (I'd argue, actually, that Lost in Translation is compelling less for its dramatisation of an unconsummated intergenerational romance than for its dramatisation of the audience's unconsummated desire to get rat-arsed in Tokyo with Bill Murray.)

There's a website, billmurraystory.com, that serves as a repository of reader-submitted first-person testimonies of Murray encounters (tagline: 'No one will ever believe you'). Some of them are true; most are basically Bill Murray fan-fiction. (Bill Murray stealing a fistful of popcorn at a benefit concert. Bill Murray strolling into a café and solving a complicated mathematical equation in a student's notebook.)

The site was set up by a Philadelphia web developer named Bill Kilkpatrick, initially as a way of indulging his and his friends' penchant for making up tales about Murray run-ins. In the early days it was all fake stories, but as it gathered momentum, he says, people started uploading stories about real encounters. I asked him whether he has any sense of what's true and what's made up. "It's weird," he said. "Half the fun is that you don't really know. Some of the ones where it's just like an unexpected kindness or an interesting exchange – yeah, I mean they're probably true. And then some of the crazier ones, it's like it could go either way."

What is the meaning of this vast repository of real and imagined narratives? Why is there no comparable phenomenon around Tom Cruise or Jennifer Lawrence or Ben Kingsley or Jay-Z or Sigourney Weaver, or any other marquee-name celebrity? What is it about Murray that has allowed him to transcend the category of mere fame to become a kind of folk hero at the centre of his own postmodern myth cycle? Obviously, people really like his film performances, and he's clearly an extremely funny and charismatic person off the screen.

But there's more to it than this. Unless some scandal or tragedy intervenes, what we ordinarily see of major celebrities is no more or less than what their PR representatives want us to. They tend to be either so aloof and gleamingly perfect as to seem hardly human (your Clooneys, your Jolies, your Beyoncés), or drug-or-booze-addled hot messes (your Gibsons, your Lohans, your Sheens). Whereas Murray looks to be both completely extraordinary and completely normal. He doesn't have any 'people'. He doesn't even have an agent. He seems to take full advantage of his own celebrity while being at the same time weirdly impervious to it. (Famously, he has an 800 number, whereby people who want to hire him for an acting job can call up and leave a message, which he'll reply to if he's interested. I contacted his lawyer, by the way, to see if he might be willing to pass on this number. No dice.)

Murray seems to be constantly doing things for no other reason than that they're fun things to do. But it's a very specific kind of fun: generously inclusive and yet somehow paradoxically self-enclosed, as though his public manifestations – the kickball games, the party crashing and so on – were a kind of in-joke with himself, a way of keeping himself amused that just happens to involve amusing the whole rest of the world.

Bill phones a friend

One of Murray's oldest friends is the writer/director Mitch Glazer, who is married to the actress Kelly Lynch. In an interview with the AVClub.com, Lynch mentioned that whenever Road House (in which she featured in some pretty explicit scenes with the late Patrick Swayze) is on TV, Murray makes a point of picking up the phone and calling Glazer. "Every time Road House is on," she said, "and he or one of his idiot brothers are watching TV – and they're always watching TV – one of them calls my husband and says, 'Kelly's having sex with Patrick Swayze right now. They're doing it. He's throwing her against the rocks'. Bill once called him from Russia."

A revelation from Bill

A woman named Anne contributed a tale to Billmurraystory.com about an encounter with Murray on a flight from Manila in the 1980s. "Close to the end of the flight," she wrote, "I asked if he would give me an autograph. He was kind enough to oblige me. Little did I know, he would give me an autograph that I would recite to people for years." She also uploaded a photo of the artefact, a

Philippine Airlines

letterheaded page with the following handwritten message: "Anne, Your dad sold black market crude rubber to the Japanese in WWII. I'm sorry. I'm very sorry. Bill Murray."

Bill goes driving

While attending the 2007 Scandinavian Masters golf tournament, Murray was stopped by Stockholm police after he was found driving a golf buggy in the city.Tournament organiser Fredrik Nilsmark said the golf cart had been on display outside the hotel in which Murray was staying when he borrowed it, seemingly to go to a nightclub. "I ended up stopping and dropping people off on the way like a bus," Murray explained. "I had about six people in the thing." While the driving of the golf buggy was not a crime, police became suspicious when they smelt alcohol on Murray's breath after pulling him over. Asked about the incident later that year, he said, "They assumed that I was drunk and I explained to them that I was a golfer".

Bill props up the bar

On billmurraystory.com a contributor called Jack recounted meeting Murray last summer. He was in a restuarant with friends when "one of the waitresses came up and told us that Bill Murray was drinking at the bar". They moved closer to get a better view. "Upon seeing him," Jack wrote, "I thought that he didn't look the greatest. In fact, he looked like he hadn't taken off any of the make-up he had on in Zombieland". Regardless, he was determined to talk to Murray. "I walked up to him and said something generic like, 'Mr Murray, I just want to say I'm a huge fan of your work'. He turned around, finished his drink, patted me on the shoulder and, in an almost whisper, said, 'Please, call me Dr Murray' and walked away."