The scenario, often termed devo max, is backed by 32%, one point ahead of independence and seven points clear of more limited devolution.
The proportion grows to about two-thirds support for devo max when people are given a straight choice for that option against either independence or the status quo.
The new details, from ScotCen's Scottish Social Attitudes survey, also suggest people are not clear about the implications of greatly extending MSPs' power.
There is a split among people wanting public services funded from Edinburgh or London, as well as on whether the rate of income tax should ever vary from the rate in England.
Devo max is not available to voters in the referendum this September, leaving it up to unionist political parties to explain how the Scottish Parliament may be given greater powers after a No vote.
Professor John Curtice, research consultant at ScotCen Social Research, said: "Both Labour and the Conservatives have to decide in the next few weeks what proposals for more devolution they wish to bring forward in advance of the independence referendum.
"Doubtless supporters of the idea will note its apparent widespread popularity, while sceptics will point to the relative lack of enthusiasm for some of its consequences. But given that as many as 44% of those whose first preference is devo max have yet to decide how to vote in September, working out the best way of appealing to their sometimes seemingly inconsistent feelings and aspirations could well prove vital in determining the eventual referendum outcome."
Bruce Crawford, an SNP MSP who chairs Holyrood's Referendum Bill Committee, said: "Westminster refused point-blank to include devo-max on the referendum ballot paper, so the only way to achieve the powers over pensions, welfare and tax which the overwhelming majority of people in Scotland want is to vote Yes for an independent Scotland."
Drew Smith, Scottish Labour's constitutional spokesman, said: "Devolution gives us the best of both worlds: a strong Scottish Parliament backed by the economic certainty and security of the United Kingdom. The only threat to devolution is Alex Salmond's referendum on breaking up Britain.
"Labour has a proud record of delivering on devolution for Scots. It was a Labour government which delivered the Scottish Parliament in 1999 , we played a central role in putting through the Scotland Act, and look forward to setting out our proposals to strengthen the Scottish Parliament next month."