While 39% favour independence, almost half (49%) back the current set up, in which Scotland remains part of the United Kingdom but with decision making in a number of areas devolved to the Scottish Parliament.
The results of the 2015 ScotCen Scottish Social Attitudes survey suggest a major shift in opinion in recent years.
In 2007, the year the SNP came to power at Holyrood, 24% favoured an independent Scotland, while 62% supported remaining in the UK with a devolved parliament.
By 2014 half (50%) favoured this option, while a third (33%) wanted Scotland to leave the UK - with the figures from the 2015 survey showing a six-point rise in support for independence over a year.
The latest survey also found that 6% of people wanted Scotland to remain in the UK but not have its own parliament, while 6% said they did not know.
The findings were released on the final day of Holyrood before the next Scottish Parliament elections, and also revealed that just over half (51%) believe that “the Scottish Parliament should make all the decisions for Scotland”.
This up by 10 points from the 2014 survey, carried out in the run-up to the independence referendum, and is also the highest total since ScotCen started asking the question in 2010.
Meanwhile, another 30% believe Holyrood should make decisions on everything apart from defence and foreign affairs - the constitutional set up dubbed by some as “devo max”.
More than two fifths (43%) of those surveyed said independence would be beneficial for Scotland’s economy - the highest total ScotCen has found since it first asked the question in 2009 - while 37% believe it would make the economy worse.
Professor John Curtice, senior research fellow at ScotCen Social Research, said: “The fact that the Scottish Parliament to be elected on May 5 will be a markedly more powerful body than its predecessors fits a public mood that, following on from the referendum, is more supportive of a powerful Scottish Parliament than ever before.”
ScotCen questioned 1,288 people between July 2015 and January 2016 for its latest research.