No Bregrets
Almost as soon as the referendum votes were counted people were asking for polling on whether people regretted their decision. There is still a certain audience who seem downright desperate to find polling showing that people do not, after all, want to leave the European Union (and, I suppose, a (slightly larger) audience who want to see polls showing they don't!). I guess this is the curse of a referendum decision that takes a couple of years to actually implement.
The most straightforward way of measuring Bregret is to ask the referendum question again - how would people vote if the referendum question was asked again now. Several polls have done that:
YouGov/Eurotrack (20th-25th Oct) Remain 44%, Leave 43%
BMG (19th-24th Oct) Remain 45%, Leave 43%
YouGov/Eurotrack (21st-22nd July) Remain 43%, Leave 44%
YouGov/Eurotrack (3rd-4th July) Remain 45%, Leave 45%
All of these suggest a very small movement towards Remain, and given Leave's lead was only four points that's enough to flip the result in a couple of cases. However, I'd be a little cautious in reading too much into the results. All of these polls are just straight "how would you vote questions" with no attempt to account for differential turnout, when at the referendum Leave voters were more liable to turnout. If you look at the actual tables for these you'll find there is very little movement between remain and leave, the shift is down to people who didn't vote in the referendum claiming that in a referendum tomorrow they would vote in favour of Remain. That's possible of course (perhaps people who assumed a Remain victory in June and didn't bother to vote, now realising their vote really would count)... but I'm rather sceptical about people saying they'd vote in an EU referendum who didn't bother to vote in the one we just had.
The approach alternative is to ask if people think it was the right decision and if they might change their vote.
Just after the referendum there was an poll by Ipsos MORI for Newsnight, which showed 43% of people thought Brexit was the right decision, 44% thought it was the wrong decision. Asked if they would change their mind in a new referendum, 1% of Remain voters said they would definitely or probably change their mind, 5% of Leave voters said they would definitely or probably change their mind. If those people all switched to the other side it would have just edged into a Remain lead.
In contrast YouGov have regularly asked if people think the decision to leave was right or wrong, and have tended to find slightly more people saying it was the right decision. The pattern of opinion is pretty consistent - movement between Remain and Leave is small and tends to cancel out, people who didn't vote at all tend to split in favour of it being the wrong decision:
YouGov/Times (11th-12th Oct) - Right to Leave 45%, Wrong to Leave 44%
YouGov/Times (13th-14th Sep) - Right to Leave 46%, Wrong to Leave 43%
YouGov/Times (30th-31st Aug) - Right to Leave 47%, Wrong to Leave 44%
YouGov/Times (22nd-23rd Aug) - Right to Leave 45%, Wrong to Leave 43%
YouGov/Times (8th-9th Aug) - Right to Leave 45%, Wrong to Leave 44%
YouGov/Times (1st-2nd Aug) - Right to Leave 46%, Wrong to Leave 42%
Finally the British Election Study asked a question on whether people regretted how they voted or not. Only 1% of people who voted Remain said they regretted their vote, but 6% of people who voted Leave said they regretted their vote. Now, saying you've some regrets doesn't necessarily mean that you wouldn't, on balance, end up doing the same. For what it's worth though, if those people who had regrets hadn't voted the result would still have been leave; if those people had voted the opposite way it would've been Remain.
Looking across the board at all this polling, there is a suggestion that public opinion may have moved very slightly towards Remain, and with only a four point lead that's enough to change the lead in some polls. However, in most cases that apparent movement isn't people changing their minds, but is down to the opinions of those people who didn't actually vote last time. That means if there was another referendum right now, if turnout was similar to June the result would probably be similar too.
My expectation is that, given time, we probably will see "Bregret", simply because Brexit is going to be tested against reality while Remain isn't. The road ahead has a lot of obstacles and some Leavers' hopes and expections will be dashed (Remainers' hopes and expectations of what would have happened if we'd stayed won't, of course, face the same collision with reality). The lead at the referendum was only 4%, so it really won't take that many people having second thoughts to flip opinion over. To those who really want to see evidence of Bregret in the polls - have a bit of patience. It will probably come in time, but the data really isn't there to support it now.