Our survey of more than 3,000 voters shows both a desire for greater equality in Britain, and some acknowledgement of change in the last decade. The Fabian Society/YouGov poll was conducted across age groups, income groups and groups of varying political intention and reveals clearly why the Conservatives and Liberal Democrats are eager to claim a commitment to this issue. Our results show unequivocally that the electorate - old, young, rich, poor, Tory or Labour - wants a fairer Britain in which vast and entrenched differences in income, in opportunity and in life chances are narrowed.

The poll carries a very clear message for Gordon Brown – the British public is overwhelmingly in favour of a fair and equal Britain. The vote for an equal chance in life for everyone was even greater than even some Fabians might expect: 96 per cent of this representative group said that it was a good idea for Britain to be a place where no child grows up in poverty; and 85 per cent said in Britain the gap between the rich and the poor should be much smaller.

However, the poll shows a desire to go further than Labour has gone so far. Overall, 46 per cent felt Britain was a fair country where every child had an equal chance in life, indicating that there is a long way to go to assuage the public's desire for a fairer society, as only 34 per cent felt that Britain had become fairer in the last ten years. On this last point the difference between Labour and Tory voters was enormously significant, with 72 per cent of Labour voters feeling the last decade had brought more equality to Britain and only 10 per cent of Tories.

The Labour Party of the last decade can be accused of doing good by stealth - its little-heralded advances in the fields of maternity and paternity leave, for example, or the work done on programmes such as Sure Start or the drive to reduce primary school class sizes for the most disadvantaged children as well as the well-off. Our polling suggests this may be the right time for Labour to remove the bushel from its light and to start reminding the electorate just who has actually been responsible for such changes.

These figures make clear why Cameroonian Conservatives moved to embrace social justice and equality issues, mouthing lines about 'narrowing the gap' before being tugged back into a more traditional line by the hard right wing. It's no longer just Labour voters that believe Britain should be a fair country where each individual has an equal chance in life – increasingly Conservatives and Liberal Democrats do too.

Voters of all political persuasions now support a Britain where the gap between social grades is narrower. Across a decade in power, Labour has successfully moved the electoral territory and equality cannot now be ignored by any party.

This is a Labour victory. No political party can ignore fairness and equality issues in the twenty-first century– not even the Conservatives. As our poll shows, not only do the public believe that Britain should be a fair country but they believe more can be done. Even broken down by party affiliation a surprising 77 per cent of Conservatives approved of cutting income inequality, compared to 82 per cent of Labour voters. Support cut across regions, ages, and social grades.

With the launch of the new equality body, the Commission on Equality and Human Rights this autumn, as well as the expected announcement of the Equality Bill in the Queen's Speech and the appointment of a new Equalities Minister, Gordon Brown has an opportunity to use this clear public support for greater equality and fairness, to his electoral advantage. But he must choose his language carefully to express that vision, while there is manifestly widespread support to tackle discrimination and create more equal opportunities and outcomes, the words 'human rights' still perversely seem to stir antagonism among a British public which often appears not to believe that human rights legislation, whether in Europe or in the UK, brings the average citizen any benefit.

Strangely, as the Fabian/YouGov poll shows, while the public feels strongly about giving Britons an equal chance in life and object to discrimination – 94 per cent of the public, for instance, said nobody should suffer discrimination because they had a disability – the idea of human rights bringing greater fairness to every Briton still does not chime with the public.

It will not be an easy task for Trevor Phillips either. The new CEHR will have to attempt to win the backing of gender and race equality activists, people with disabilities, gay men and lesbians, and some faith groups for its 'joined up' approach to dis-crimination and equality.

Both the Prime Minister and the CEHR will want to harness the high levels of support for a fairer Britain as they drive forward, while tackling high expectations about further improvements to the state of the nation. Just 2 per cent thinks 'a lot of progress' has been achieved over the last ten years in narrowing the gap between the rich and the poor, although 39 per cent said at least a little or some progress had made a difference. Public perceptions are more positive when it comes to indicating if progress has been made in giving every full-time worker a basic living wage. Around 77 per cent felt there had been some level of improvement in this area, compared to 13 per cent who felt there had been no progress. Despite strong Conservative opposi- tion to minimum wage legislation, even voters of the right now vehemently support giving workers a fair wage for a fair day's work, with 94% of Tory voters supporting it, compared to 97% of Labour voters.

This is a victory for the government in terms of public opinion. And there are others. For instance, 50 per cent of the public felt progress had been made on giving children growing up in low-income families the same chances as children growing up in high-income families, while Labour voters were even more positive with 68 per cent indicating progress had been made in this area, and of the age groups younger voters were the most positive with 52 per cent believing that progress had been made. Giving every child access to a good state school also registered a good public response, with 55 per cent believing they had seen improvement; and Labour's work on the NHS showed the public believed things had got better: the response showed 51 per cent felt immediate access to high quality healthcare had improved in the last decade.

The worst result for the Government was on crime, a subject registering a welter of headlines through the summer. In our poll only 30per cent of the public felt progress had been made towards the objective of everyone being able to be out and about in their neighbourhood without fear of being robbed or attacked in the last decade. Public perception of how the government is tackling crime will obviously be a significant hurdle for Gordon Brown during this parliamentary term.

Some very high figures show just how far the case has been won on fairness. The survey found 95 per cent believed no pensioner should live in poverty, and 94 per cent felt children growing up in low-income families should have the same chances as children growing up in high-income families. When broken down by political affiliation the beliefs were interesting, 91 per cent of Conservatives backed giving low-income children those same opportunities as richer kids, only slightly less than Labour voters (96 per cent) and Liberal Democrats (98 per cent).

Other indicators also showed near-total support for the basics of fairness and equal opportunities for every single person in Britain. An overwhelming 95 per cent backed giving access to high quality healthcare to everyone: strong support came from voters across the parties, and in different age groups and different income levels. The same strong backing for making sure every child had access to a good local state school (97 per cent).

So what does this mean? It is clear that Britain as a whole now believes strongly that a fair society is vital for its future. This snapshot of public opinion shows that equality matters – to everyone.

The three major parties in the UK's political landscape were formed on differing principles to represent the differing interests of particular groups in society. Labour's tradition throughout the century of the party's existence has staked a claim to the ideal of equality, and, with the arguable exception of Lloyd George's pension system, every major legislative advance towards social, racial and sexual equality has been introduced by the Labour party.

But in a post-ideological 21st century where parties increasingly fight over the swing voters who inhabit the middle ground, the Conservatives and the Liberal Democrats are launching increasingly deep forays into the territory Labour has long claimed as its own. David Cameron, in particular, has sought to impress middle- ground voters with a language of fairness and equality that would once have sounded hollow in the mouth of a Tory leader. One of the key insights behind Blair's triple electoral success was that the party needed to appeal beyond its core and to take voters away from the centre- right. However, the very success of his strategy provides a key lesson that the other Parties show every sign of having learnt all too well for Labour's comfort. Therefore, one of the central challenges facing Gordon Brown's new government is how it ensures that the electorate remains convinced that equality remains not just a defining Labour value but a goal that no other party is truly committed to pursuing.

As the Tories seek to triangulate their commitment to wealth and hierarchy with a desire for the votes of those for whom the higher peaks of both remain chimerical, and as the Liberal Democrats seek to square their own circle of liberal economics and paternalistic concern for the disadvantaged, Labour needs not to fear encroachment into its long-established hinterland, but to continue driving its deep desire for fairness, equality and social, political and economic improvement for the many not the few. Finally the government must make sure the public takes note of what it has achieved and what it continues to plan to achieve greater equality.

• This polling was done as part of the Fabian Equality project and will be included in full in the forthcoming Fabian pamphlet 'The Equality Challenge' by Tom Hampson and Rachael Jolley