A small semi on 39 Amersham Road, Romford, has a place in British political history. It was there in 1980 that prime minister Margaret Thatcher first handed over the keys to a council tenant, sparking the right-to-buy revolution. In the following years, as Havering sold off more than 10,000 council houses, its waiting list spiralled, hitting 12,000 two years ago. But now, in a move endorsed by the coalition government, it has slashed its list by three-quarters, in a bureaucratic sweep of the pen that is being repeated in councils across Britain.

The London Borough of Hammersmith's housing waiting list has also fallen precipitately – from 8,171 people in April 2012 to just 768 in April 2013. Bournemouth's is down from 9,425 to 3,177. Warrington's has fallen from 12,091 to 3,173. Havering's is now below 3,000.

Behind the extraordinary collapse in waiting lists is not a sudden surge in the supply of council homes. Instead, local authorities are writing to applicants telling them that they no longer stand a chance of obtaining a home from the council unless they meet new local criteria.

Hammersmith has imposed a rule that people must live in the area for at least five years before they can apply for a council house. Havering, which used to allow people to apply for a council home if they had lived in the area for six months, has changed that to two years.

"We felt that it was more honest to tell people that they had no chance, and should concentrate their efforts on finding private rented accommodation, rather than bidding pointlessly for properties that they would not be able to get," said Lesley Kelly, the Conservative councillor with responsibility for housing in the borough. Other changes excluded households who were able to live with their family.

At the end of 2013, the government gave councils statutory guidance on waiting lists, making it clear that it believed they should be looking at applicants' links with an area. The guidance said: "The Secretary of State believes that including a residency requirement is appropriate and strongly encourages all housing authorities to adopt such an approach. The Secretary of State believes that a reasonable period of residency would be at least two years."

The housing minister, Kris Hopkins, told the Guardian that the new guidance was "to ensure that people in need of social housing with a long-standing connection to the area, or who have served in the armed forces, are prioritised".

The move has been contested by the Local Government Association, while housing charity Shelter says that decisions to restrict lists underline the need for more homes. "There are any number of reasons why you might need a council home," says Roger Harding, Shelter's director of communications. "It could be a disability leaving you unable to work, or high private rents outstripping your wages. The fact that councils need to choose between people who typically all have fair reasons for being on the waiting list, highlights the urgent need to build more affordable homes."

But in Havering, councillor Kelly rejects the idea that there is a problem with the supply of homes. "My theory is we have got enough housing in Havering, we've just got the wrong people living in the wrong homes," she says.

Under-occupation of large homes by older residents is one of the main problems the council has identified. The area has an older population than the rest of London, and the council believes some are in homes better suited to families, and is trying to entice them to move. Downsizing would be a win-win says Kelly. "They can't heat it, they can't maintain it, the neighbour complains that the garden is not being looked after," she says.

Rather than begin a major new home-building programme, Havering prefers infilling existing plots and ensuring the mix of homes is right. It is spending £5.5m on new homes, topped up with £2m from the Greater London Authority. The car park and a patch of grass next to an existing sheltered housing development has been identified as a location for three two-bedroom bungalows. Low-rise blocks are being extended to add one- and two-bedroom flats, and lock-up garages razed to create space for small developments of family homes. It also aims to demolish small 'butterfly bungalows' which take up space but offer inadequate bedsit-style housing for older people, and to replace them with larger family homes.

"Lots of schemes, like using the garage sites, are quite radical," says Kelly. "Other councils are watching to see how we get on."

Applicants who are being scrubbed off waiting lists are being expected to find homes in the private rented sector. But in common with landlords around the country, many in the borough are not willing to take on tenants who receive housing benefit. Hilbery Chaplin, an agency in the town's market square, is one. Lettings manager Kim Berry says: "It's the way housing benefit is paid – four-weekly, while we charge rent for a calendar month; it is paid in arrears, while rents are paid in advance; and it goes to the tenant rather than the landlord. Any property that comes on gets snapped up very quickly; landlords have the choice so they will go for the professional tenants."

Meanwhile, new rules intended to revive the right to buy council homes – which give tenants discounts of up to £100,000 – mean that Havering's council housing stock continues to shrink. Over the past 12 months, it has sold off 100 homes – 30 more than it has plans to build.

How to get a council house

• Some councils outsource their lists to a housing association but the council website should still be your first port of call.

• Each council sets its own rules; you may not be able to join if you don't have a connection with the area, earn too much or do not earn at all.

• You will need to back up your application with lots of documents, including ID, details of where you live now, and proof of your connection to the area if necessary.

• You can register for more than one list (assuming you meet the criteria).

• Councils use a points or banding system. Priority is usually given if you have serious health problems, a home that is very overcrowded or lacks basic facilities, or if you are at risk of domestic abuse.

• If you are homeless or about to lose your home you may be prioritised.

• If you were seriously injured in the armed forces you should get priority.

• You may lose points if you have been involved in anti-social behaviour or have a history of missed rent payments.

• Shelter has a guide to obtaining council housing at shelter.org.uk.

• Some councils prioritise people with jobs, or people who are volunteering.