A New York singing superstar is not the type of customer Anna Evans usually serves at the till of the Co-op in Shawlands, Glasgow.

So when Lana Del Rey – beloved by fans around the world for her hits Video Games and Born to Die – strolled in looking for bread rolls, Evans was understandably suspicious.

"When she came to the checkout I said 'you look like Lana Del Rey' and she said 'I am'," said Evans, 48.

To prove who she was Del Rey pointed to a copy of Vogue, featuring the singer.

Evans asked for her autograph: "She said 'I'll sign it for you in the photo shoot part of the Vogue magazine'," she says. "So she did and I've got it to this day." Did she pay? "She offered. She gave me five pounds to pay for it, but I said no."

Chance encounters with America's hottest indie export in Glasgow are becoming increasingly common. Recent sightings include her sipping a pint in Oran Mor, browsing vintage clothes on Byres Road and rushing through Central Station with a Mulberry bag in tow.

The 26-year-old singer is to appear at Glasgow's SECC on Thursday, and demand for tickets has been so high that two nights at a smaller venue were merged and moved to the huge arena.

Del Rey's love affair with Glasgow stems from her relationship with Kassidy guitarist Barrie-James O'Neill, who lives in the city. She has been visiting his Scots hometown on and off for years – well before she became a recognisable face in Shawlands and the west end – and now visits the city three or four times a year.

She has also adopted her boyfriend's favourite football club, Celtic, and has tweeted a photo of herself in a Celtic top. When asked last January by Esquire magazine where her permanent home was, Del Rey responded: "I don't have one ... [But] when I have any time, I go to Glasgow. That's where I'm happiest. I found this band that I love and I live with them out there."

Another fan who saw the star in Glasgow was Gillian Rae, 22, who works in Vintage Guru on Byres Road. "I was actually on my lunch upstairs when I heard she was in and I bolted downstairs to see her. I never bothered introducing myself or anything. I was too embarrassed. She's beautiful," Rae said.

After flicking through the racks for around half an hour with her boyfriend in tow, Del Rey picked out a natural fur coat. "She was worried if she had enough money on her card. We made a joke about that," Rae recalled. Pictures of the singer wearing the coat later triggered criticism from animal rights groups.

"She was in conversation with her boyfriend a lot. They seemed quite in love. That was nice. She was asking his opinion on the clothes." Earlier this year media reports suggested the pair were engaged.

One Glasgow fan, Niamh Mackay, 17, who is off to the SECC gig on Thursday, said: "She's a goddess – an inspiration. I only live a few miles from Shawlands and I can't believe she was there. It's a holy site of pilgrimage now. I will be stalking the Co-op this week in the hope of seeing her ... if I get enough revision for my Highers done."

Back at the Co-op in Shawlands, Anna Evans still regrets not being able to talk to the star for longer. "She just stood at the end of the checkout wanting to gab to me. Unfortunately, it was 6pm and I was serving a long queue. Everyone was huffing and puffing, all wanting to get served," she said.

"She would have stood at the end of the checkout talking for ever to me. She was so nice. She came around the other side of the check out, hugged me, gave me a kiss on the cheek and said 'I'll pop back in and see you'."

If Del Rey does drop back in, Evans will be ready: "I'll take her upstairs for a cup of tea."