Sort; figure out. But also address in a more general sense; solve (as in a headline from the Daily Mirror: “Sort Out Those Sleep Demons–For Good!”). I just went to nytimes.com and was greeted with a banner advertisement (from Ameritrade) announcing, A LOT OF FACTORS IMPACT YOUR RETIREMENT. WE’LL HELP YOU SORT THEM OUT, which reminded me that I have been meaning to look into this phrasal verb.
Brits seem to have a fondness for many sorts of sort. There is Harry Potter’s sorting hat, of course; the noun roughly equivalent to type, as in a decent sort; and, notably, the qualifier sort of, which not only is preferred to the U.S. kind of but seem to be taken to in a more existential way, as a contingent and unenthusiastic attitude toward life. U2 has a song called “A Sort of Homecoming,” and one of Graham Greene’s memoirs is (brilliantly) titled A Sort of Life. In Tom Stoppard’s play The Real Thing, a pop-music-besotted character refers to once being taken to Covent Garden to hear “a sort of foreign musical with no dancing”–about as good a definition of opera as I have come across. (In fairness, I should also note that the very American William Carlos Williams has a poem called “A Sort of a Song.”)
I have the impression that sort of is getting some legs in the U.S., especially in academic circles. However, I am here today to talk about sort out. Neither Google Ngram nor Google Trends shows any great increase in American use, though Ngram suggests that in Britain, figure out is a not one-off Americanism, having overtaken sort out in about 1995 and maintaining a substantial lead. Even so, I am going to invoke blogger’s privilege and claim sort out as a not one-off Britishism.
Laura Hoptman, the curator of the exhibition, which runs until January 6th, has sorted out a quiet but potent development of the last decade by focussing on an international array of twenty-six young but established artists of many tendencies. (Peter Schejeldahl, The New Yorker, November 4, 2002)/Mobile phone carriers, banks, credit card issuers, payment networks and technology companies are all vying to control these wallets. But first, they need to sort out what role each will play and how each will get paid. (New York Times, March 23, 2011)
notoneoffbritishisms.com 
Southerner here. It’s not uncommon to hear a phrase like “sort things out” meaning “solve problems,” in American English, I think. Your British headline’s usage sounds sort of funny to me, but the Ameritrade one sounds fine.
Ben – there’s also the recentish (10-15 years) expression ‘sorted’ to mean fixed or to signify job/deal done. Colloq. wide boy street chat, often one word so: “Did you get the money for that old car?” Response. “Sorted.”
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I’ve heard fictional UK toughs use “sort you out” as a metaphor for beat the daylights out of you.
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I wonder, though, if it is more against Britishisms entering the American lexicon, rather than merely observing their evolution in discourse.
Thanks K. As for your question, I am going to weigh in soon on the observe/disapprove issue. Watch this space.
I work with people in the UK and am familiar with their expressions and phrasing. What I have heard them say is “sort”, not “sort out” e.g., “We’ll get it sorted.”
“Sort out” is a familiar American expression, not a UK one.
Thanks, Paula. Do native UK-ers agree that “sort out” is not used there?
“We’ll get it sorted” is very common. “I’ll sort it out” or “I’ll sort that out” certainly don’t sound anachronistic to me, but my English parents (in their 50s) would use that form, whereas I and most people I’ve come across, here in Scotland at least, would leave off the “out”, thus “I’ll sort it” or “I’ll sort that”. I believe it’s considered gramatically incorrect, but it may indeed be the most common usage these days (amongst us trendy young people, anyway).
Google News search for “sort out” in U.K. sources yields 690 hits in March and April 2011, including this from today’s Evening Standard: “[Andrew] Marr claims that this piece of jiggery-pokery allowed his family to sort out a private matter in private.”
As a Brit, I agree that “sort out” is widely used either to make arrangements for something to be done, or to “work out” (meaning “figure out” or “solve”) a problem/solution.
“Sorted!” is a cheery expression for “Job Done!” or “Ta-Dah!” but also has a drug context, as in “I know a bloke who’ll get you sorted for tonight” or the 1990’s song lyric “Sorted out for Es and whizz” (i.e. well supplied with ecstasy and speed/amphetamines)
It is my experience that the use of sort which remains most exclusively British is noted by K, Paula and Dynamic. My Brit pals from Yorkshire use it exclusively instead of “done” as well as a threat or commentary on being taught a phyisical lesson, as in “right, I’ll sort you out, proper, you pillock, if you pinch my bum again,”
A good few years ago we went skiing in Vail. I volunteered to find somewhere to stay. The accommodation(s!) office was run by an absolutely gorgeous piece of American womanhood-married,sadly.
An Englisman comes in and says “sorted now, thanks” and disappears.
The baffled look on the face of the GPOAW told me she required an explanation, so I took an inordinately long time to tell her that this was Brit shorthand for “thanks for trying to find somewhere for me, but I’ve found a place myself”.
Sadly, I never saw her again.
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My friend used to say “Just sort it out” to me when I had troubles and I took it to mean “get your act together girl!”