United Kingdom

The EU regards the four freedoms (of goods, capital, services and labour) as sacrosanct and non-negotiable for single market membership.

"Freedom of movement" of people within the EU can legally be restricted even under the current rules to avoid benefit shopping etc., and mechanisms to avoid undercutting local wages (one of the Lexit arguments) would not be hard to impose.

By having bought into the right-wing tropes on these issues, the two have boxed themselves into the same corner, and they'd lose face without some fancy footwork to explain any change of heart. What was a general, vague hand-waving vote to "Leave the EU" has been transformed into "of course that means leaving the customs union/single market, that's what the people voted for" even though we have leading Brexiteers on record during the referendum as insisting it didn't mean any such thing.

An about-face would be most embarrassing for May, who as Home Secretary will be best remembered for her "hostile environment" while she failed to enforce the systems allowable under EU rules which would have restricted residency for those moving from other EU countries who failed to find work within a certain period, and would have imposed conditions on access to healthcare etc. The only explanation for why she didn't do so (other than the usual incompetence - she failed to reduce immigration despite all her efforts) is that the government knows that EU immigrants bring net financial benefits (among others) and imposing such a system would require expenditure on a bureaucracy to track them.

I can't see how a solution can be found that avoids no deal while she's still prime minister, and time's fast running out to replace her (I'm also not hopeful that Corbyn would win a general election, let alone grasp the opportunities to change tack with hardliners like Seamus Milne, O'Donnell and Unite leader McCluskey in the picture).

For those scared of and opposed to immigrants, a no deal Brexit won't be a solution, as those they're most likely to object to don't even come from the EU, and we're going to need more of them to make up for the shortages of EU workers.

Leaving the EU customs union is what will cause the major problems at the UK's borders - in Dover and Northern Ireland etc.

The much-vaunted "Norway-with-as-many-pluses-as-you-like" deals for access to the single market but not the customs union would require accepting the four freedoms and also making a financial contribution (Norway pays around £140 per capita per year), and wouldn't avoid the need for border checks.

A "Canada"-style agreement for limited access to the customs union wouldn't entail accepting the four freedoms, but would exclude services - the backbone of the current UK economy (Turkey and some smaller countries also have some access to the customs union, but not for services).

Any trade agreement with the EU is likely to include restrictions on the UK's terms of trade with non-EU countries. Any standalone trade agreements we eventually manage to make with non-EU countries would likely include quotas for visas enabling their citizens to move and work here. It's also not hard to see how a trade deal with the US, for instance (especially if Trump or his like remain in power) could compromise UK standards and restrict options in other trade deals.

Any of these options would involve the UK becoming a rule-taker, not a rule-maker, inflaming the neo-sovereigntists, and negotiations with anyone - the EU or other prospective trade partners - would likely take years while the UK's bargaining position and economy progressively weakened.

The only hopeful signs recently are the efforts by backbenchers from both the main parties and others to sidestep their leadership and reach across the aisles to seek agreement, though it's probably too little, too late now - which is the major recurrent theme of this whole sorry exercise.

And this is just the general agreement in principle phase. If the UK doesn't just crash out of the EU at the end of March, we have the nitty-gritty detailed negotiations to look forward to!