I would like to know which spelling is more common in the UK: fantasise or fantasize?
My 1983 Chambers gives the -ise spelling first, which seemed reasonable to me because I think the -ize one looks 'American'.
But my 1998 Chambers gives the -ize spelling first, so I guess I have to assume they've changed their tune in the interim. Note that this change is reflected for all relevant words in the later edition. As this article points out, almost all British 'authorities' apart from OED (Oxford English Dictionary) have now adopted/endorsed the American standard.
For what it's worth, NGram suggests that ever since the word[s] really started to take off in the 60s, fantasize has been more common even in British English. I think maybe the fact that it was so rarely used before has allowed Brits to take the 'easier' path of simply copying American usage; there's less historical usage to be swept under the carpet, and it seems to have been more used by Americans earlier on anyway, so it's only fair they get to say how we spell it.
Thanks to @Peter Shor for commenting that the 'original' British form was phantasise, as can be seen from this NGram. In light of that, perhaps the 'pseudo-original' fantasise could be seen as an abortive attempt to mimic (a dying) 'standard' British spelling for historical reasons. Which failed/is failing because there simply isn't enough history to sustain that position.
The spelling fantasise is originally British; and appears to be more prevalent in British literature than fantasize (34 instances of the former for every 20 of the latter) according to the British National Corpus. (Thanks, @z7sg!)
Google NGrams is misleading here, showing that it is not more common than fantasize in British literature; but since the database is not entirely accurate, its evidence must go disregarded in this case.
"-ise" is commoner than -"ize" in British English, certainly. But the answers that assume that -ize is purely an invasion by American are barking up the wrong tree; it has a long and honourable history in Britain. My 1965 edition of Fowler has a long article, quoting the OED, which concludes that verbs derived from Greek (the vast majority of -ize/ise words) should follow the English -ize rather than the French -ise. (It also includes the sentence "But the Oxford University Press, The Cambridge University Press, The Times, and American usage, in all of which -ize is the accepted form, carry authority enough to outweigh superior numbers": ah, the good old days.)
And there was an Inspector Morse mystery (though I forget the name), turning on the fact that an Oxford lecturer would never have written a note conataining 'organise' and 'realise'.
Both 'fantasise' and 'fantasize' are equally correct in Standard British English. However, only 'fantasize' is permitted in standard American usage. The '-ize' suffix is not an Americanism, not nonstandard, and is more etymologically correct — that is, it is the older variant and closer to the original Greek. It is, however, less common in British English today, probably because it's perceived to be an Americanism.
Whatever spelling variant you use you ought to use it consistently throughout any given document.