Labour is battling to avoid a disastrous third place in next week's election, according to the latest Guardian/ICM poll. It suggests Labour support – 28% in today's poll – has not yet hit bedrock and could fall below 20%.

The ICM figures show the Liberal Democrats' surge is sustained, with all three party scores identical to last week's Guardian/ICM poll. The Conservatives retain a narrow three-point lead on 33% ahead of the Lib Dems on 30%. Labour is close behind on 28%.

Compared with the most recent ICM poll, published on Sunday, Conservative support is down two, Labour up two and the Lib Dems down one – but those movements are small and all results are within the margin of error. Support for other parties is eight points, with no sign of an advance by the Greens or Ukip.

Conservative support is at the same level as the party achieved in the 2005 general election and would leave the party far off a majority. That will add to anxiety within the Tory campaign.

On today's figures, Labour could theoretically become the largest party in parliament despite coming third in votes. In practice it is likely to lose more seats than uniform national swing suggests.

Today's poll shows Conservative support is holding firm, with 83% of supporters saying they do not expect to change their mind before polling day.

By contrast, only 69% of Lib Dems and 68% of Labour voters say they will stick with their current choice. Almost a third of people supporting Labour say they might end up backing another party instead.

If that happens, Labour's vote would fall to a minimum of 18%. Its maximum potential vote – all current supporters plus people who say they may decide to back it – is 33%. That is three points lower than the party achieved in 2005.

By contrast, the Tory minimum is 27%, and maximum 37% – a level which would probably give the party a small majority.

The Lib Dems vote is the most volatile: the party's minimum current support is 23% and its maximum 40% – which if achieved would give the party first place in votes if not seats.

Some people who say they support the Lib Dems also say they are far from certain to vote, and some did not vote at the last election. Figures for party support are weighted to reflect the likelihood of each party's supporters actually turning out on the day.

Nick Clegg is thriving as the party leader least tainted by spin and most associated with political change. But many voters are sceptical of his ability to lead a government and see voting Lib Dem as a protest against the other parties.

Overall, 58% agree with the statement "the Liberal Democrats represent something new and different", against 37% who disagree. That includes 56% of current Labour supporters and 39% of Conservatives.

The Lib Dems are appealing most of all to young voters – 64% of 18-24s think the party is different. Pensioners are the least impressed, with only 49% agreeing.

Meanwhile, 40% of all voters say they can imagine a Liberal Democrat government led by Nick Clegg as prime minister, while 55% say they cannot.

Lib Dem supporters overwhelmingly see Clegg as a potential PM: 80% say he could run the country. But only 16% of Conservatives and 31% of Labour supporters agree.

However voters reject the idea that the Lib Dems are only a protest party: 51% reject the idea.

There is also evidence that the bandwagon effect is helping the party pick up votes: 31% of all voters, including 26% of Labour supporters, say knowing that other people are switching to the Lib Dems encourages them to do the same.

However Clegg's party lags behind on policy. On the economy, the subject of this week's final leaders' debate, the Tory team of David Cameron and George Osborne are narrowly ahead of Labour's team of Brown and Alistair Darling, 30% to 29%. Clegg and Vince Cable are third, picked by 23%. Among Liberal Democrats' supporters, 21% think the Labour team is better placed to run the economy.

That tight race is reflected in the battle over cuts: 50% say spending should start to fall now (up four since March) while 44% say it should not (down four). Supporters of cuts now narrowly outnumber opponents, but only Conservative voters are in favour. Most Labour and Lib Dem voters do not want immediate cuts, suggesting Cameron may not pick up support if he repeats his call for them in the debate.

Finally, there is some sign Cameron is recovering support: 38% say he is campaigning best for people like them, up five on last week. That is nine points ahead of Brown but nine behind Clegg, who leads on 38%.

Meanwhile 35% say Cameron is the most competent potential prime minister (up three), against 29% who say Brown and 22% Clegg.

The Lib Dem leader is overwhelmingly seen as the most honest leader: only 17% say he is more spin than substance (up four on last week) against 24% who say Brown (down five) and 44% who say Cameron, unchanged.

ICM Research interviewed a random sample of 1031 adults aged 18+ by telephone on 23-25 April 2010. Interviews were conducted across the country and the results have been weighted to the profile of all adults. ICM is a member of the British Polling Council and abides by its rules.