Data from 107 local authorities shows 86,000 households have been forced to look for one-bedroom homes, of which only 33,000 have become available in the past year.
The figures mask considerable regional variation. In Essex, 100 social housing tenants in Rochford were deemed to require a one-bedroom property because of the benefit changes but only five had become vacant the previous year.
In Gloucester the council said 111 one-bed homes had been available last year, but almost 500 households needed them because of the bedroom tax.
Inverclyde in Scotland said 1,100 households would need to move into one-bedroom homes – of which just 96 had been free to rent last year.
False Economy, the trade union-backed campaign that used freedom of information requests to get the data, said it had chosen to focus on one-bed properties as ministers had been forced to acknowledge last year that there was a "shortage" of such homes but pressed on regardless with the policy.
A spokesman for False Economy said: "The disparity between the demand for one-bed housing and a whole year's worth of supply is so severe that there is little hope of plugging the shortfall."
Without new homes being built, "tens of thousands are now facing a crisis".
Homeless charities also called for the policy to be abandoned. Leslie Morphy, chief executive of Crisis, said: "Without enough one-bedroom homes to move into, tens of thousands are powerless to avoid the anxiety, debt and arrears caused by the bedroom tax.
Our fear is that many, through no fault of their own, will in the end become homeless as a direct result of government policy.
Ministers must accept these facts and rethink the bedroom tax now."
Labour described the policy as the "worst combination of cruelty and incompetence". Liam Byrne, the party's spokesman on welfare, said the bedroom tax was a "mess". "Thousands of vulnerable households are trapped by this hated tax with no option to move