Various varieties of Brexit crop up in the British media, but it has all started to get a bit unmanageable. So here, as a sort of summary, are twenty varieties of Brexit!

Soft Brexit
One of the two original varieties of Brexit, Soft Brexit means the UK would leave the EU but retain the closest economic ties with it. This would mean either staying within the Single Market or the Customs Union, or both. This Brexit variant would have a relatively minor economic impact on the UK. It can also be known as the Norway model. With a Soft Brexit the UK would still have to retain Freedom of Movement of people, pay into the EU budget, and possibly still accept the judgments of the European Court of Justice.

Hard Brexit
The other original variety of Brexit. Here Britain would leave the Single Market (and presumably also the Customs Union as well) and would negotiate a trade deal with the EU. UK-EU relations would then be akin to those between Canada and the EU. The UK would then also be able to have more control over immigration and would not pay into the EU budget. However this variant could have a major negative economic impact on the UK.

Very Hard Brexit
A more extreme version of Hard Brexit where instead of striking a trade deal with the EU, the UK would seek to use WTO rules for trade instead. While the UK is a member of the WTO it does not have its own Schedules and these would have to be negotiated – not a simple task. If even this failed an Ultra Hard Brexit would ensue, where the rules for the UK’s trade with everyone else would the same as non-WTO members’ trade with WTO members. Any Very Hard or Ultra Hard variety would only be likely if negotiations between the UK and the EU somehow failed during the negotiation period foreseen by Article 50 of the Treaty of Lisbon.

Train Crash Brexit
First coined by Gideon Rachman in the FT, this version is where the UK and the EU realise very soon after Article 50 is triggered that the negotiations are going nowhere, and both sides retrench into opposition to each other. This ultimately leads to a Very Hard Brexit after the two years foreseen in the Article 50 process.

Cliff Edge Brexit or Dirty Brexit
This is a variant of Train Crash Brexit, but where the problems only really come to a head towards the end of the 2 year period in Article 50. Britain would then crash out of the EU without a plan as to how that would work, with the UK possibly reverting back to WTO rules for its trade with the EU (a Very Hard Brexit). Politico has called this a Dirty Brexit, but I think Cliff Edge Brexit is more appropriate to describe it.
(thanks Jack Bailey (@SocSciStatsGuy) for Cliff Edge, Tom Phillips (@flashboy) for Dirty Brexit)

Chaotic Brexit or Dog’s Brexit
Both sides take a messy and unpredictable approach to the Brexit negotiations, and as a result the negotiations get prolonged beyond the two year deadline foreseen in Article 50. Negotiations drag on and flip between being tense and cooperative, but do not cross the line into Train Crash or Cliff Edge as everyone realises the dangers of those. Jeremy Corbyn has also used the words Chaotic Tory Brexit, but I take this more as an attack on May than as a serious Brexit variety.
(thanks Andy Ellis (@ndls61) and Richard Elwes (@RichardElwes) for the Tory aspect)

Clean Brexit or Smooth Brexit
The UK government talked about this at the time the Miller case first went to the High Court back in October, arguing that a clean and orderly timetable for Brexit was imperative. Clean here means the opposite of Dirty, and does not mean a clean break. To achieve this it assumes that both sides negotiate in good faith and that each side has enough resources to conduct a smooth and straightforward negotiation. This strikes me as wishful thinking. Philip Hammond has also advocated a Smooth Brexit, trying to soothe banks with the idea of a Transitional Brexit (see below).
(thanks Anna Gumbau (@annagumbau) for Clean, and Das V (@dasvee) for Smooth)

Red, White and Blue Brexit or Bespoke Brexit
The UK government has refused to be drawn on what sort of Brexit variety it wants, saying it will aim for a bespoke deal as the Norwegian model, Swiss model and Canadian model are not to its liking. This could be Britain’s superiority complex, perhaps why Theresa May called for a Red, White and Blue Brexit – which we presume refers to the UK’s Union Flag. In reality any bespoke deal would include elements of the relationships other non-EU countries have with the UK.

Grey Brexit
Middle ground between Soft Brexit and Hard Brexit – and a way for Theresa May to keep her cabinet together, as reported by Politico. I am not sure whether this one is any different to a Red, White and Blue Brexit, but it sounds a whole lot more boring and depressing. So maybe we ought to use this term more.
(thanks James Chalmers (@ProfChalmers))

Transitional Brexit
Many politicians and experts have argued that concluding Brexit within the two years foreseen in Article 50 in the EU’s treaties is unrealistic, and hence a transitional deal is needed. Here the UK starts out with some sort of Soft Brexit arrangement – probably staying in the Single Market and the Customs Union but leaving the institutions – and then the negotiations start to determine the UK’s long term relationship with the EU. This is probably the most realistic Brexit scenario, but it is loathed by hard core Brexiteers who see it as backsliding on the core commitment to leaving the EU.

Have Cake and Eat It Brexit
This is where the UK proposes to stay in the EU Single Market and control freedom of movement and even possibly refuses to pay into the EU budget or be subject to the European Court of Justice – i.e. a combination of demands that are going to be refused by the rest of the EU. Demanding this in the short term will almost certainly lead to a Train Crash or Cliff Edge Brexit in the medium term.
(thanks Alison McCormick (@alijmcc))

Brexit-in-name-only or Schrödinger’s Brexit*
Brexit that formally happens (in that Britain legally leaves the EU), but in practice, especially economically, it is as if Britain has not left. This could happen if a Transitional Brexit actually eventually became permanent, or if Britain decided it wanted to stay in the Single Market and the Customs Union. It is unlikely this variety would gain favour within the UK government.
(thanks Constantine Fraser (@camfraser) for In-name-only, and Dan O’Brien (@Danopedia) for Schrödinger’s)

Reverse Brexit
An extremely cunning Brexit variant, most eloquently explained by Steve McCauley. Rather than the UK leaving the EU, and causing all sorts of problems with Northern Ireland and Scotland when it does so, here England and Wales leave the UK, leaving the rest of the UK within the EU. No Article 50 needed, and the Scottish independence issue is solved as well. Simple!

Unlawful Brexit
Theresa May’s determined Brexit course has rather marginalised UKIP and its new leader Paul Nuttall. So he is now advocating the swiftest and hardest Brexit possible – by simply repealing the European Communities Act and leaving the EU immediately, and not needing to use Article 50. This would lead to the hardest of hard Brexits, and is probably unlawful.
(thanks Christopher Huggins (@chris_huggins))

Smart Brexit
This one is advocated by Belgian PM Charles Michel, with the emphasis on good economic relations between the UK and the rest of the EU. How that actually differs from a Soft Brexit I am not sure, and I doubt even Michel knows. I presume he thinks he sounds smart by putting Smart in front of Brexit?
(thanks Anna Gumbau (@annagumbau))

Lexit or Progexit
Lefty Brexit varieties that see the European Union as an evil liberalising mendacious capitalist plot, and wish to turn the UK into a socialist panacea when Britain leaves. This of course conveniently neglects the fact that Theresa May is the UK Prime Minister. Progexit is a slightly less extreme version where social justice and equality are somehow to be prioritised as the UK exits.
(thanks Ben Judah (@b_judah))

Lixit
Liberal-Brexit. A Brexit variety where the Brexit vote is not interpreted as UK shutting itself off from the world, but indeed the opposite – where a liberal and tolerant UK trades openly and freely with the world, and where liberal values are prioritised internally within the UK.
(thanks Aaron Ellis (@AaronHEllis))

Flexit
The variety advocated by the likes of Richard North and Roland Smith, summarised by Bloomberg here. This is essentially a Soft Brexit variant that looks closely at the way Norway-EU relations work, and emphasises the emergency migration brake that is foreseen in the EEA agreement. Its advocates acknowledge it would be a hard slog to make it happen, but it is one of the more coherent and consistent Brexit varieties.

Titanic Brexit
Brexit will be a “Titanic success” said Boris Johnson in November. No-one quite knew if he meant Brexit will be a huge success, or if Brexit would go down the same way as the ship named Titanic, coming a cropper on the EU’s iceberg. I am not sure this is a serious Brexit variety, although Brexit failing awfully could indeed happen – see Cliff Edge Brexit above.
(thanks Jamie St Leger (@tweetyjsl))

Crème brûlée Brexit
This is a new contribution. Brexit needs to be hard on the outside – for the sake of how it is presented to the UK media and the population, and yet soft underneath – for the sake of the economy and British business. Just like a crème brûlée. Hence the name. Let’s make sure this one catches on!
(thanks DK Mutley (@mutley6969uk) for the excellent and amusing suggestion!)

* – there is some debate about the meaning of Schrödinger’s Brexit. I took it to mean that the UK was both in and out of the EU simultaneously. But Richard Elwes argues it’s more Hard and Soft Brexit simultaneously. Which makes it more like the crème brûlée variety!

[UPDATE 20.12.16, 1240]
An honourable extra mention for Wrexit that I had otherwise forgotten – that Brexit wrecks the EU, and hence there is nothing really to be a part of as the EU collapses. Thanks @htfb for reminding me of that one.

[UPDATE 20.12.16, 1340]
A friend on Facebook pointed out my description of Very Hard Brexit was inaccurate due to the way it explained the WTO aspect of that. The text above has hence been updated to make that all clearer.