Councils in England are to face cuts of almost 10% next year in their core central government funding.
The coalition said the "formula grant" from Whitehall would be reduced by 9.9% in 2011/2012 and by 7.3% in 2012/2013.
The cuts are part of a policy to cut central funding to local authorities by 28% over four years.
Ministers promised a "new democratic settlement", with more money being found to help deprived areas. But Labour called the cuts "devastating".
In the House of Commons, Communities Secretary Eric Pickles said councils would see their "spending power" reduced by on average 4.4% next year, with no authority facing a decline of more than 8.9% for any of the next three years.
These figures - published for every council - are smaller than the overall 9.9% because they include all other grants and income that councils get, such as council tax and NHS funding.
- Giving local people and organisations the right to buy community assets like shops, pubs and libraries; if a council decides to sell a property, community organisations will get extra time to develop their bid
- Allowing communities to question how services - such as children's centres, care homes and transport - are being run, and potentially take them over
- More power for local people to overrule planning decisions, decide where new homes should go and protect green spaces
- Powers to create directly elected mayors in 12 cities
- Powers for people to approve or veto "excessive" council tax rises
The bill also includes plans for a change in the role that councils play in finding accommodation for homeless families.
Instead of being obliged to house families who are eligible, councils would be able to discharge their responsibilities by finding them private rented accommodation for at least 12 months.
In a statement, Mr Pickles told MPs: "This will be a progressive settlement, ensuring fairness between different parts of the country."
He added that £650m would be set aside so every council could freeze council tax without hitting local services. The government would provide those who froze council tax with the equivalent of a 2.5% increase in funding.
To help protect front-line budgets, £200m would be provided to help councils modernise and cut back office costs, Mr Pickles promised.
He said: "Taxpayers are no longer prepared to write a blank cheque for the public sector. But they do want less interference in their local communities from Whitehall government.
"So the coalition government is delivering the most significant shift in power from officials in London to elected local councils in a generation."
For Labour, shadow communities secretary Caroline Flint told the BBC that councils had been told to expect cuts over four years, which were being "front-loaded", rather than being spread more evenly over the period.
She added: "You are imposing unprecedented cuts on town halls the length and breadth of the country.
"Today we find out what the government really plans to devolve to local councils - the most devastating cuts in funding for a generation and the blame for difficult decisions."
Baroness Easton, chairman of the Local Government Association, said: "We have to face the fact that this level of grant reduction will inevitably lead to cuts in services.
"The government has recognised the impact the cuts will have on those areas of the country that rely most heavily on the public sector and has provided a limited amount of new money to help those areas cope.
"However, it still remains the case that the cuts are front-loaded rather than spread evenly across the four years. Councils now face incredibly tough choices about the services they continue to provide and those they will have to cut."
Tony Travers, from the London School of Economics, said: "There's been nothing like this in modern times. If you look at, for example, Denis Healey's efforts in the late '70s to cut public spending, it had a one or two-year impact on public expenditure - including on councils - but nothing like this. "
He said councils estimated 140,000 jobs could go by the end of the process - but conceded that figure had been disputed by Mr Pickles.
The Localism Bill includes plans to make councillors approve and publish pay rules for their chief executives, powers to create directly elected mayors in 12 cities and powers for people to approve or veto "excessive" council tax rises.
The government says its "community right to buy" will give local groups legal rights to name assets like shops, pubs, libraries and leisure centres on a council "most wanted list" - if it is put up for sale, local people will be given time to prepare a business plan and raise the funds they need to bid for it.
There will also be a "community right to challenge" - giving community groups, parish councils and others the power to challenge and take over a local service. The government says this could include running children's centres, social care services and local transport links. Councils would be obliged to respond to the proposal and if they turn it down, they must publish the reasons.
Paul Brant, Labour deputy leader of Liverpool City Council, told BBC Radio 4's World at One: "Some of these reported powers are illusory. The government is making savage cuts to local government finance... Without the money, some services will just fall off the edge."
But Stephen Greenhalgh, Conservative leader of Hammersmith and Fulham Council, said: "We have to recognise and make the case that we can get by on less money. I think with less money you become more creative about how you tackle things."