Millions of people face losing libraries, sports centres and museums if "significant" further cuts are made to council budgets, a report says.
The Local Government Association says potholes could go unfilled and street lights could be switched off, with funding for some services down by 90%.
It is warning the government against making more reductions in the 2015-16 public spending review.
But the coalition called the warning "shrill and alarmist".
The Local Government Association (LGA), which represents council leaders, has "mapped out" the likely impact of a 10% funding cut to county councils and unitary authorities in England in 2015-16.
It predicts that, on average, they would have to save £30m on top of the reductions already made.
This, it argues, would mean reducing spending on a "broad combination of non-statutory services which might include children's centres, museums, libraries and sports centres, as well as reduce road maintenance budgets, increase bus fares and switch off streetlights between midnight and dawn".
The LGA says that "almost all of councils' money would have to be spent on explicit statutory responsibilities like social services, waste collection and concessionary travel, meaning that the money available for all other services, such as libraries, road maintenance and leisure facilities would have been cut by 90%".
There is a risk councils "will no longer be in a position to provide key services, which are every bit as important to the public as those provided by emergency services, the NHS and schools", it adds.
The LGA wants the removal of ring-fencing of health and schools budgets, which could, in part, "be much more effectively spent on council services such as pre-hospital care for the elderly and infirm, or early intervention services for troubled children and their families".
Chairman Sir Merrick Cockell said: "In many council areas we have now reached a stage where noticeable cuts to local services are a mathematical certainty unless the next spending round places local government finance on a sustainable footing.
"The government has to take steps which deliver long-term efficiency across the whole public sector and encourage government agencies in local areas to work together to find ways of improving services and making savings."
But Local Government Minister Brandon Lewis said: "Every bit of the public sector needs to help pay off the deficit by the last administration.
"The LGA's shrill and alarmist claims lack credibility, given councils are losing £2bn a year from uncollected council tax, £2bn from fraud and are sitting on £16bn of reserves. The LGA's own research shows that since 2010, residents' satisfaction with their councils has actually gone up, despite the need to make savings.
"Rather than predicting the end of the world, the LGA would be better to help councils deliver the savings from improved procurement, joint working and better property management."