Do you mean non-English as in a foreign language or non-English as not from the country that forms the southeastern part of Great Britain?
I’ll limit myself to the bands with recent releases on the pop/rock axis and not go to world music.
Sigur Ros, Bjork, and Gusgus from Iceland. Sigur Ros sings in a made up language, and Bjork does the occasional Icelandic song.
The Hives are from Sweden and rock out. But they sing in English, so this disqualifies them if you are talking about English language.
Since England does not include Wales, Ireland or Scotland, you can list a whole helluva lot from there. In terms of this year’s music I’d list Franz Ferdinand (Scotland), Belle & Sebastian (Scotland) and The Thrills (Ireland). Jem is from Wales, and I quite like her new disc. And of course, Tom Jones… 
But of course they all sing in English. If you are looking for people singing in another language, Sinead has done some stuff in Celtic.
From Japan, I’d list Pizzicato Five and Cibo Matto (really Japanese women living in New York). Cibo Matto is in English, but just barely so.
There’s this organic electronic disc I have by a Japanese group called Rovo which has also impressed me terribly (kind of in the Tortoise klangfarben style but more electronic). Another electrolounge group that has really got a good thing going is Royksopp from Norway. Check out their disc “Melody AM.” Mostly non-vocal, but they do have some English lyrics.
From France, I’d go with Air (who are almost painfully French) and Stereolab (which is Anglo-French). And of course the grandfather of cheeze electrolounge, Dimitri from Paris. Une Very Stylish Fille, indeed. All sing in French and English (around 50/50, with Dimitri using samples mostly in French and English).
From Mexico, I’ll go with Volovan. Great Mexican mellow pop.