I have checked back to my references for crystal clarity, referring to the New Oxford Style Manual (2012) and to the Chicago Manual of Style, although these do not state every instance and are only conventions adopted, or not adopted, by publishers. There are also instances of exceptions so these are far from being strict rules.
When introducing a phrase at the end of a sentence
A phrase at the end of a sentence can be introduced with an em dash (closed up) and the example given in New Hart’s Rules which, in my copy of the New Oxford Style Manual is p81 (4.11.2)
Everyone understands what is serious—and what is not
So, this is a single parenthetical usage and replaces a colon and is widely accepted in UK English.
When used as a pair of dashes
Where you use em dashes with a space before and after, these would be parenthetical. This is not currently a general UK English publisher’s preference.
New Oxford Style Manual (2012) 4.11 p72 states:
‘The en dash … Many British publishers use an en dash with space either side as a parenthetical dash, but Oxford and most US publishers use an em dash.’ And
(p80) ‘The em dash …Oxford and most US publishers use a closed-up em dash as a parenthetical dash; other British publishers use the en dash with space either side. …
A pair of dashes expresses a more pronounced break in sentence structure than commas, and draws more attention to the enclosed phrase than brackets:
‘The party lasted—we knew it would!—far longer than planned’
… Use an em dash spaced to indicate the omission of a word, and closed up to indicate the omission of part of a word:
‘We were approaching — when the Earl of C— disappeared’
So, both the em and en dashes can be used to set off an augmenting or explanatory word or phrase in a sentence that could stand alone without the insertion. Examples include:
That small flower – the pink one – is as fresh as can be.
That small flower—the pink one—is as sweet as can be.
He knew the price of that rare vintage – everyone did.
She knew the price of that rare vintage—everyone did.
In the UK, it is conventional to use a SPACED EN DASH. This is not a law but is the conventional style used my many (although not all) UK publishers (an exception being Oxford).
In the US, it is conventional to use a CLOSED-UP EM DASH. Not a law just a convention, but one which many US publishers follow.
It is also true that there are some style guides which ask for spaced em dashes, however this is relatively rare.
So how are authors using the dash conventions?
Beyond the reference material it can be useful to look at how published authors’ work appears.
Publishing in the UK
In Stephen King’s Everything’s Eventual (2002)
There is an example of the single parenthetical en dash:
(p299) ‘… for what I’ve done – for what I did to Skipper, even.’
And in Val McDermid’s Killing the Shadows, (2001)
(p159) ‘That someone hated Kit – or his word – even enough to pour out such venom…’
Whereas publishing in the US
In Emma Donoghue’s Room, (2010) there is an example of a pair of parenthetical em dashes:
(p133) ‘Old Nick will carry you into the hospital, and the first doctor you see—or nurse, whatever—you shout …’
And in David Baldacci’s The Finisher (2014) there is an example of the single spaced em dash:
(p1) At first light, I was almost always up in my tree — a stonking, straight-to-the-sky poplar with a full towering canopy.
In conclusion: I recommend using spaced en dashes or closed-up em dashes because that is what readers are most familiar with. The choice of which depends on where much of the target audience are (US or UK) however where there is an international audience either style can be chosen, as long as it is applied consistently.