Most of the people riding electric scooters around Copenhagen glide like swans, bodies motionless and serene with heads tilted into the sunshine.

I am the ugly duckling, wobbling and jerking while I struggle to figure out exactly how this two-wheeled contraption works. Where do I put my feet? How hard do I thumb the accelerator? And how on earth am I meant to indicate left and press the brake with my left hand at the same time?

It doesn’t take long to start gliding, even if I don’t feel very swanlike, or to notice that people in Denmark have found all sorts of less conventional ways to use e-scooters since they were introduced in January. Couple after couple perch together, heels of their feet barely balancing on the footplate. A traveller presses his suitcase between his knees while he crosses the junction near Tivoli Gardens. Two teenagers dart along the footpath outside the city’s central station, nipping between pedestrians.

“I’ve seen a guy sitting on a suitcase using it as a scooter chair,” said Jesper Hemmingsen, a researcher at the Danish council for road safety, the RST. “I’ve seen parents delivering their children to school, a child standing at the front and their parents steering. It’s an easy way to get there, but it’s not a safe way. I’ve seen parents hire scooters for their kids to give them something to do. People treat them like toys. But they have no protection.”

Over the past 18 months, it feels as though e-scooters have taken over many of the world’s cities. Scooter share firms have staked claims in more than 100 cities worldwide. Lime and Bird (from the US) and Voi, Circ, Flash and Tier (from Europe), are start-ups dripping with venture capital backing, attempting to create and carve up a market that is changing cityscapes everywhere.

But piles of scooters discarded by the roadside, a worrying number of injuries and even some road deaths have provoked a growing backlash in some places.

Those dangers were underscored recently when Emily Hartridge, a 35-year-old YouTuber and TV presenter, was killed while riding an e-scooter in Battersea, south-west London, after a collision with a lorry at a roundabout. Last month Paris – the e-scooter hub of Europe, with an estimated 20,000 trottinettes on the streets – saw its first fatality after a young man was hit by a truck.

So are e-scooters a vital new part of modern, eco-friendly urban transport, or are they a risky and unnecessary fad, pushed by tech investors desperate to disrupt the status quo? And what can people in the UK, where the use of e-scooters remains illegal on both roads and pavements – despite the rise in models for sale in shops – learn from the experience of other European cities?

Copenhagen, where e-scooters arrived nearly six months ago, is well-placed to answer this question. The police and road safety council are assessing the impact of the seven scooter firms and 7,000 scooters that have arrived since the Danish government introduced a trial period on 17 January.

Pedestrians complain about scooters strewn across pavements and that they are a hazard for blind people and wheelchair users, as well as other anti-social behaviour: one young man rode an e-scooter through the aisles of a Netto supermarket in Odense seven times in one day.

Walking around the Danish capital with Hemmingsen, it’s clear that the city’s road network is much more friendly to e-scooter riders than London. The reason is the Danes’ love of cycling. “In Denmark you’re born, you’re potty trained, then you cycle,” Hemmingsen said. “Copenhagen is the second most bicycled city after Amsterdam – we cycle 1.4 million km every day.” Only 3% of people here don’t ride bikes every day.

The city is flat, which helps, but more importantly there are miles and miles of wide roads with dedicated cycle lanes, usually raised and separated from car traffic with a kerb. No one has to compete for space. A scooter shouldn’t need to be anywhere near a lorry.

But the road safety council is still worried about a growing number of injuries. “At the moment we have an assumption that it’s the same level of injuries for e-scooters as for cyclists,” Hemmingsen said, mentioning a recent study by two casualty departments in Copenhagen that found an average of about two injuries a day. About 800 cyclists a year are seriously injured.

Copenhagen police are also acting. After running an information campaign, senior officers mounted an operation against drink-riders earlier this month, posting officers around the city centre.

In one night last weekend, 24 e-scooter riders were stopped on suspicion of riding while over the legal alcohol limit, and four more for some form of narcotic intoxication, superintendent Allan Teddy Wadsworth-Hansen said. They face fines of up to 2,000 krone (£240).

One of the motorcycle officers standing in Frederiksborggade, a main route into the heart of Copenhagen’s night life, stopped one rider and breathalysed him. Then a second arrived who had also been drinking. “While he’s still trying to process the first one, a third e-scooter turns up with two people on it,” Wadsworth-Hansen said. “When they see the police, they try to slow down. The woman on the back falls off – she wasn’t hurt seriously – and the driver turns out to be driving under the influence of alcohol. And then a fourth one approaches, and he has also been drinking.”

Although there is sometimes a spike in the number of car drink-drivers over the summer, “you would never get four drivers in a row over the limit,” Wadsworth-Hansen said. “It doesn’t really bode well for e-scooters late in the evening, because drinking and driving is dangerous.”

“Road users are used to seeing bicycles, but they’re not used to e-scooters. When you see someone standing up, arms by their sides, it’s easy at a glance to think they are a pedestrian and not realise they’re coming towards you at 20km/h,” he said.

The experience of Parisians suggests that it’s not only e-scooter riders at risk. Isabelle van Brabant was hit by an e-scooter in the park at Les Halles in May, breaking her wrist in two places and causing a bone tear. For the pianist at the Paris Opera, it’s a professional disaster as well as a personal one. “She cannot play piano for one year, and piano is her life,” said her husband Jean-René Albertin. “When Isabelle was at the hospital, they said that each day 10 people come in for treatment. We think about 200 people a day are hurt.”

They have set up a campaign group, Apacauvi, to win compensation for victims, with legal actions planned against the Paris municipal authorities and the rider who injured Van Brabant. “We have received 10,000 messages and we have about 850 people who want to participate in the association – people who have been hurt,” Albertin said.

He wants the French government to force riders to take out insurance before they start riding, so that victims can be compensated from a central fund, even if an individual rider cannot be identified.

Paris’s mayor, Anne Hidalgo, was in favour of e-scooters as a greener alternative to cars, but she has announced a crackdown on the “anarchic” dockless scooters. From this month, bad parking, riding on pavements and breaking the 20km/h speed limit will all result in fines. Police said they had issued 1,000 tickets and impounded 600 scooters.

Albertin is unconvinced. “People just ignore the rules,” he said. But he is not in favour of an outright ban, as in the UK. “I don’t think making them illegal is right.”

Britain remains a lacuna for the e-scooter share firms because, for at least the last 30 years, it has been against the law to ride them either on the road or the pavement. Electric scooters, like Segways, hoverboards, Go-Peds and self-propelled unicycles, are neither roadworthy enough under the Road Traffic Act 1988 to be registered as vehicles – complete with licence plate – nor can they be used on the pavement: an offence against the Highways Act 1835. People are occasionally prosecuted.

Yet the e-scooter trade is flourishing. Halfords has 15 models available in the “kids zone” on its website. Argos has 23, also marketed as toys with designs from Trolls, Star Wars and Jurassic World. Ministers are in a bind, under pressure to liberalise or be left behind, while also wanting to enforce the law to prevent further tragedies like the one in Battersea.

Shadow transport secretary Andy McDonald told the Observer there was a “pressing need for the government to provide clarity and guidance” on e-scooters.

In March, then transport minister Jesse Norman told the Observer he was considering licensing e-scooters, and his successor Michael Ellis is overseeing a review of legislation at the Department for Transport. But the government is also concerned about retailers flouting the law, and Ellis will hold a roundtable with Halfords, Lime, Bird and some manufacturers to urge them to tell consumers that they might be able to buy an e-scooter, but they can’t ride it in a public place.

“We are examining whether they can be used safely on the road – and if so, how that should be regulated to ensure the public’s safety,” Ellis said. “However, companies must understand that reviewing laws does not necessarily mean laws will change. People who use e-scooters need to be aware it is currently illegal to ride them on the pavement and the road.”

At night, the streets of Copenhagen are busy with e-scooters, and it’s easy to see why so many people have been stopped recently for drink-riding. You’re free to travel on a whim: it is cheaper and more immediate than cabs, far more convenient than buses or trains, and doesn’t require the effort of riding a bicycle.

“It has completely divided people,” Hemmingsen said, as we walk round the meatpacking district. “Either people think they are the best thing ever or they are a scourge. I saw someone had put one in a rubbish bin. They dredged the harbour in March and they found 12, and scooters have only been here since January.”

Here, there are neat rows of five Lime scooters on the street corners, and we soon catch up with the person leaving them there: Andrew, a Romanian driving a Lime van who has been working in Copenhagen for two months, repairing and recharging the scooters.

He shows me how the handles, the GPS tracker and other parts of the scooter are damaged – “by drunk people” – but not very often. In other cities, scooter-share companies will pay freelancers, nicknamed “juicers”, a bounty for each scooter they charge, but Andrew says that there are 20 people like him in Copenhagen who are all well-paid Lime employees.

There is some dispute about how long an e-scooter will last – some need to be replaced after just 28 days’ use in Paris, Albertin says, while Hemmingsen says they can go for up to three months. Andrew says many last much longer. The closest comparison in the UK is with dockless bike companies, three of whom have pulled out of the market within a year, complaining that vandalism in Britain is higher than anywhere else they have done business.

The e-scooter firms say they take safety very seriously, and each app will remind riders of some basic safety tips when they scan and hire a scooter. Kristina Nilsson of Voi said the Swedish firm was in the process of creating dedicated parking spots in the 31 cities where they operate, giving riders the incentive of extra credits if they leave them in the right place.

“We’re also in the process of rolling out speed zones, so in certain areas you go slower than 20km/h,” she said. Nilsson claims the crackdown by Paris is having an effect: “I understand Paris will have a tender for having only three players there. And we hope to be one of those three.”

Some cycling charities see e-scooters as potential allies in the battle to get better cycle paths. Rachel White, head of policy at Sustrans in the UK, said they were “generally supportive” of scooters, if speeds were limited and they were kept off pavements. Beyond safety, they are only worried that having access to effortless travel might make us put in less effort. “A slight concern is that they don’t have the same health benefits as walking or cycling,” she said. “But generally, another group of organisations lobbying for more protective cycleways can only be a good thing.”

The UK’s road network may yet dictate the future of e-scooters. In London the volume of traffic dwarfs the number of cars in Copenhagen while the cycling network is far less developed. London has 239km of cycle paths – less than the 375km in Copenhagen, and serving 13 times as many people. And outside the capital, the fragmented and often unreliable public transport system means commuters are wedded to their cars.

Paul Hodgkin, CEO of UK mobility start-up Ginger, believes that in spite of this, Britain can introduce e-scooters safely. His firm is in talks about putting e-scooters in business parks and campus universities – on private land. “Innovation is happening elsewhere. Wait-and-see is not a neutral position. If we take too long, the industry will move on and we won’t be able to help set the model and get it right,” Hodgkin said.

• This article was amended on 15 July 2019 to clarify a comparison of cycle paths serving London and Copenhagen, measured in kilometres.

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