'I have never protested about anything before but I feel like this is a really important cause," said Rebecca Anderson, 23, standing in Glasgow's George Square yesterday.

She was one of thousands who took to the streets to show their opposition to the UK Government's bedroom tax, which will take effect from tomorrow.

Anderson added: "I've read about how many disabled and vulnerable people this affects and I felt I really wanted to show my support against it. It's so obviously wrong."

Just how many people joined the protest is subject to the usual hotly-argued debate. Official police estimates of the march from Glasgow Green to George Square put the figure between 2000 and 3000, with another 1000 staging a separate protest in Edinburgh. Protesters in Glasgow say the figure was higher – some put it as high as 10,000. Whatever the exact turnout, it was a sign of the deep unpopularity of the benefit cut, with similar protests all over Britain yesterday.

The most hated of the cuts – and the one which critics of the Government believe has the potential to become the "new poll tax" – will reduce housing benefit paid to those living in social housing who are deemed to have a spare bedroom.

Protesters argue this will disproportionately affect disabled people and women, and could lead to evictions, homelessness and even deaths.

Fiona Jordan, chairwoman of the West of Scotland Federation Against the Bedroom Tax, said the group was pleased with the turnout in Glasgow.

She said: "We are delighted at the number of people who have come out to stand together and say there will be no evictions because there will be no bedroom tax. We will not allow this policy to come into force."

Karl Fletcher, 41, also on the march, said: "It's an odious policy, really terrible, and its affects are far-reaching. I'm down from Aberdeen for the weekend with my family but I wanted to come along and bring the kids along to give support to this."

Everywhere you looked at the protests there were T-shirts and placards with the word "ratbag" superimposed over the face of Iain Duncan Smith, the Westminster Work and Pensions Secretary who had once seemed to symbolise a more caring brand of Toryism after a much-publicised visit to Easterhouse.

His reputation these days is different – the "ratbag" insult was hurled at him by furious protesters during a visit to Edinburgh last week.

"He is a ratbag," said Kelly Parry, representing the National Union of Students, yesterday.

"We need to fight back so that never again will the people of this country be robbed by Westminster."

Bill Scott, from disability campaign group Inclusion Scotland, said two-thirds of UK properties affected house a disabled person, rising to four-fifths in Scotland.

"The purpose of an equality impact assessment is to find out and then change your policy so it doesn't disproportionately impact on the vulnerable," he said.

"Instead, the UK Government went ahead and implemented that policy completely unchanged. It is a disgrace."

Alice Bowman, who campaigns for the Radical Scottish Independence Campaign, said: "Only four out 45 Scottish MPs voted in favour of the bedroom tax.

"In an independent Scotland, we will have our voices heard, and have control of our own budget."

A large number of Yes Scotland banners were paraded alongside banners from the pro-independence Scottish Green Party.

But Bowman was heckled by one protester, who said: "Alex Salmond could stop this tomorrow and he's not doing anything about it."

The SNP did show its support for the marchers, in the form of a statement from SNP MSP Linda Fabiani.

It said: "The Scottish Government has brought in mitigation measures and has made a firm commitment to scrap the policy in an independent Scotland.

"SNP-led councils have committed to not evicting tenants affected by the bedroom tax, in contrast to several Labour local authorities who have refused to follow suit.

"There is no doubt the bedroom tax is an abhorrent policy and the level of outrage surrounding what is being imposed is entirely justified."

Socialist politician Tommy Sheridan also turned out to lend support to the campaign in Glasgow.

He said: "This policy has prompted meetings to spring up across the west coast of Scotland and has brought people together with a determination to fight that I have not seen since the demonstrations against the poll tax.

"This is not about left and right, it is about right and wrong and the bedroom tax is wrong."