Ukip has fallen back from the giddying heights it reached after May's council elections, although the party remains a force to be reckoned with – as does the anti-political mood it represents.

That is the message of a new Guardian/ICM poll, which shows that Ukip has slipped back by a substantial six points from last month's record-breaking level of support – leaving it on 12%.

The poll shows that Labour is ahead on 36% (up two on the month) and the Conservatives on 29% (up one).

ICM poll June 2013

Nevertheless Ukip's 12% is three points above its previous high point in the Guardian/ICM series until last month.

That leaves Nigel Farage's party tied with the Lib Dems, who inch up a single point from last month's nadir of 11%. The combined score of other smaller parties is 10%, made up of 2% for the Greens and a buoyant 8% for the combined Scottish and Welsh nationalists.

On the all-important question of handling the slump, both the major parties are losing trust in parallel. David Cameron and George Osborne maintain a nine-point lead over Ed Miliband and Ed Balls as the most trusted team for economic management, but only because faith in the Labour duo is waning almost as fast.

In December last year, when voters were asked to put aside party preference and concentrate on who could manage the economy properly, 35% preferred Cameron and Osborne and 24% Balls and Miliband. Today those respective figures are 28% and 19%. A growing band – now a majority of 52% – either say "neither", "don't know" or refuse to answer.

Labour has never polled well on economic competence since Gordon Brown lost power in 2010, but – after a week which saw both Balls and Miliband make big speeches about the need to contain elements of public spending – the party will be bitterly disappointed to have fallen below 20% on this score for the first time with ICM.

Its only comfort is that trust in the Tories has fallen even more sharply, at a time when the coalition's top brass is quietly hoping it can see green financial shoots.

Graph showing voters' view of the economic competence of the Labour and Tory leaderships.

At the start of last year, 46% trusted Cameron and Osborne with the economy, a proportion which had already fallen to 35% by last December, and which has now fallen again – to sink below 30% for the first time.

Distaste for the political mainstream is also evident in increasingly miserable personal ratings for both Cameron and Miliband.

Months of Conservative infighting have started to register with the voters. The perception of unity used to be a major strength for Cameron; as recently as the end of last year, voters believed he had the backing of his party, by 62% to 29%. Today, only 44% accept that suggestion while 45% disagree, making party division a net negative for Cameron for the first time with ICM, even if only marginally.

Miliband fares much better on this count – judged to be backed by his party by 65% to 20%, a 45-point gap in his favour, which builds on the already-impressive balance of +43 that he scored at the end of last year, and the somewhat smaller positive scores for unity that he notched up earlier in his leadership.

On other characteristics, however, the Labour leader polls much less well. One closely watched measure of leaders' standing is voters' faith in their being "good in a crisis".

In December 2011, David Cameron scored well on this count – 50% said he was good in a crisis and only 40% said the reverse. Even after the omnishambolic year that followed, he was still rated as good in a crisis by 48% in December 2012, compared with 44% who thought he wasn't.

But today, despite Cameron's recent dispatch of statesmanlike duties in the aftermath of the murder of Lee Rigby in Woolwich, only 37% of voters say his crisis-handling skills are good, while 52% think they are not, giving a net negative of -15.

By contrast Miliband has always struggled on this test of leadership mettle – by double-digit margins voters have always refused to believe he is good in a crisis – and his position is not improving.

His 21%-44% negative score on this count in December 2011 had narrowed somewhat to 28%-45% by the end of 2012, but opinion is now leaning against him more heavily again on this score – by 21%-49%.

The 28-point gap between those two figures is not quite a record (he did even worse in April 2012), but a perceived lack of resolve in emergencies is nonetheless a huge impediment for an aspiring prime minister.

A big polling negative for the old Etonian prime minister has long been a perceived gulf with ordinary people, but today he sinks to new depths on this score – only 29% now think he "understands people like me", as opposed to 65% who disagree. Miliband has tended to do much less badly on this empathy question, but by 46% to 41% voters also say he fails to understand them, a five-point gap that represents a modest worsening of his position on this score since the end of last year, when the balance against him was only 46%-44%.

As well as showing declining respect for Britain's chief political leaders, the poll charts remarkable stability in the race between them – Labour's seven-point edge over the Conservatives is very much in line with the recent average in the ICM series, where – since last October – the lead has stood at an average of 7.5 points.

That is a somewhat smaller gap than in some other polls, largely because of adjustments that ICM makes for so-called "partial refusers" – respondents who reveal how they voted in 2010, but not their plans for next time.

In Tuesday's poll, ICM's adjustments make a substantial difference to the headline results (reducing a 12-point Labour lead to just seven) because of the record high numbers of such "shy voters", who represent 15% of the sample.

The split of the partial refusers is fairly typical – the biggest chunk, 39%, are "shy Tories", a third – 33% – are "shy Lib Dems" and only 19% are "shy Labour"; when ICM applies assumptions that half of such voters "return home" to the party that they backed at the last election this works to reduce the Labour lead.

ICM's Martin Boon explained: "With voters plainly fed up with politicians of all stripes, it seems that fewer are ready to admit – perhaps even to themselves – who they will plump for next time. That leaves us with more adjusting to do than normal."

ICM believes such adjustments are central to the accuracy of modern opinion polls. Its adjustment procedures date back to 1992 – the last general election that the polls got spectacularly wrong – and it has increased accuracy in successive general elections since.

• ICM Research interviewed a random sample of 1,002 adults aged 18+ by telephone on 7-9 June 2013. 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.