Nine of those claims have come from users in Auckland with the rest in Christchurch.
Fletcher said she wanted an urgent report on the scooters and the safety issues related to their use on crossings, footpaths and roads.
"I am quite shaken by this."
Auckland Mayor Phil Goff believed there was a serious issue, and wanted a safety report ahead of any major accident. "They are becoming very popular, very quickly."
He said he had discussed the issue with the Minister of Transport Phil Twyford this morning.
His concern was prompted by a visit to the Viaduct, where he saw a huge number of riders zooming around.
"Amazed really at the speed at which people were travelling on something with very small wheels ... without the requirement of wearing helmets and with the ability to travel on the footpath," Goff said.
"We do need to examine whether regulation of scooter use is adequate from a safety perspective.
"I'd like to get an early report, rather than waiting after the event and responding to an accident ... I don't want to see the idea of using scooters stillborn, I think they offer another option for travelling around the city."
Goff said he would request a council CEO memo by the end of next week: "The intention of this is not to stop the trials nor the scooters. I am quite favourable [towards them].
"Let's look at what we need to do to consider sensible safety considerations. This is about safety."
Councillor John Walker supported the trial of the Lime e-scooters and said he did not want Auckland to become a nanny state. He was almost hit by a car this morning.
Goff said the scooters had been on his mind: "I don't want to exclude the use of scooters... but there are issues. I am not sure if we are capable of regulating it through bylaws."
Goff pointed out discrepancies between cyclists, who needed to wear helmets, and scooter riders - travelling at up to 27km/h - who did not.
Councillor Richard Hills said he was worried about demonising scooter riders, but urged context around other transport issues including 30km/h speed limits and more cycleways.
"I have been in Elliott St and groups of kids are racing each other," said councillor Linda Cooper.
Other councillors said there were already enough bylaws, including public nuisance bylaws. Another bylaw was not needed.
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• Eli Orzessek: Why Lime electric scooters are an accident waiting to happen
Transport Minister responds
Transport Minister Phil Twyford said e-scooters offered a cheap, quick way to cover short distances and connect with public transport: "We are always balancing the need for mobility with people's safety and scooters are not different," he said.
"Part of the solution will be creating more space in our street for people walking, cycling and using low-powered vehicles like e-scooters."
The Government was already planning to invest $390 million over the next three years with councils to expand footpaths, shared paths and cycleways.
"The Government is also looking at whether we have the right level of regulation for low-powered vehicles. This include looking at potential speed limits and what vehicles should be allowed on the footpath.
"Councils already have the ability to make bylaws that control how footpaths are used. Many councils already regulate devices like skateboards and kick-scooters."
'Someone is going to die'
The sudden proliferation of e-scooters has drawn stark warnings from two safety advocates, both of whom want a number of changes to road rules.
"Someone is going to die if we don't see changes," workplace safety campaigner Lance Wiggs says.
Veteran motoring writer Clive Matthew-Wilson said: "I predict multiple injuries and probable deaths in the near future".
"E-scooter riders are allowed to ride without helmets and share the footpath with vulnerable pedestrians, such as old people and young children. Inevitably there are going to be horrific collisions."
On September 24 there was the first reported e-scooter death in the US. An SUV hit 20-year-old Carlos Sanchez-Martin as he rode a rented Lime e-scooter along a Washington DC street.
According to the Associated Press, the SUV dragged Sanchez-Martin about 20 yards, pinning him beneath the vehicle. He died at a hospital shortly afterwards.
A Lime rider also died on September 1 in Dallas, though the exact cause of his death is not clear.
A class action lawsuit has just been filed in LA, where it is alleged careless practices by e-scooter ride-sharing companies have led to multiple injuries.
Here, there has been a crash on Auckland's Tamaki Dr, requiring an X-ray after cuts and bruises were suffered. ACC says it has fielded 14 e-scooter claims in the eight days since Lime was launched in NZ.
Matthew-Wilson says he is 'shocked' that the government has allowed e-scooters to be used without much thought for the consequences.