Insurers are counting the cost of the storm but say it is too early to tell whether it will compare with the multibillion-pound hits caused by previous severe weather events.
Initial estimates of the level of financial damage wrought are not expected until later this week.
But the Association of British Insurers (ABI) says the great storm of 1987 cost around £2 billion in today's money while the summer floods of 2007 resulted in a hit of more than £3 billion.
Insurance giant Aviva said it already had teams on the ground helping those affected by the latest storm while extra staff were drafted in to deal with customer calls overnight and during the day.
The South East and the South West experienced the most problems, Aviva said, with damage caused by falling trees and walls, lost tiles and weather damage to roof areas the most common claims so far.
ABI is urging anyone who has suffered damage due to the storm to contact their insurer as soon as possible for advice.
Aidan Kerr, head of property at the ABI, said: "Events like this are exactly what insurance is for.
"Insurers understand the devastation that weather like this causes and their first priority will be to work as quickly as possible to deal with claims and help customers recover."
The ABI said those making a claim for damage should have all necessary insurance documents to hand when contacting insurers. Most have 24-hour helplines and will talk people through the process, the body said.
It said insurers would try to deal with customers as quickly as possible, arranging temporary accommodation and emergency payments where necessary and assessing damage when safe to do so.
The single storm event is unlikely on its own to cause rises across the board in premiums but those making claims could find themselves losing no-claims discounts they have built up, the ABI said.
Other advice included keeping receipts when arranging repairs to be carried out to stop any damage worsening.
Those suffering flood damage were told to check it is safe before using electricity, gas and water supplies.
They were advised to disinfect floors and furnishings once flood water recedes and, where practical, to leave doors and windows open. Where possible, rooms should be kept heated.
Home owners should not rush to redecorate as it can take months for a property to dry out fully.
Rob Townend, director of claims for Aviva, said: "The high winds have caused significant damage to property which will be a new and difficult experience for many people.
"So we know it is important to explain to people how it all works, what they can expect and what it means for their home and their family.
"And with storm damage there is no such thing as an "Act of God" exclusion - this is exactly what your insurance covers you for.
"So any damage caused to your property as a direct result of storm force wind and flood water is covered including damage caused by falling trees and if you are forced to move out of your home the cost of alternative accommodation is also paid for.
"We will make emergency payments where necessary and we can make separate arrangements for your pets too."
However, Aviva said most companies expected homes to be in reasonable condition so wear and tear was not covered while fences, gates and hedges that have blown down as a direct result of high winds were also generally excluded.
AA Insurance said calls for car insurance claims were about 20% higher than normal, with claims ranging from falling trees to scaffolding collapse.
AA Insurance took twice as many calls for home insurance claims compared with a normal autumn Monday, but added claim sizes so far are generally not as big as major storms in the past.
Simon Douglas, director of AA Insurance, said it was too early to assess the cost of damage but home insurance claims were generally for damaged roofs or trees falling on buildings or out-houses.
He said: "Claims are generally in the order of hundreds of pounds, rather than thousands, although some flood claims could be much greater."
And he said the storm is unlikely to reverse a trend for falling home insurance premiums.
"I believe that it would take a very severe winter to reverse the downward trend in the short term," he said.