In American English, cozy is the standard spelling of the word meaning, mainly, (1) comfortable and warm, and (2) to make oneself comfortable and warm.1 Outside North America the word is spelled cosy. Both spellings are used in Canada, though cozy is more common in published writing from this century. These spelling preferences extend to all derivative words, including cosier—cozier, cosying—cozying, and cosies—cozies.
The word has taken many spellings through its history in English (it’s thought to derive from Gaelic origins and entered English in the 17th century)—including cosey, cosie, and cozey in addition to the modern spellings—and neither of the forms now preferred gained permanent ascendancy until the 19th century.
Americans use cozy:
The case has been seen as offering a window on the cozy relationship between lawmakers and hospitals. [New York Times]
A similarly creepy trope in children’s literature is the narrator who cozies up to the child reader. [Wall Street Journal]
Jannings would strike his own deal with the devil, starring in pro-Nazi films and cozying up to the Third Reich. [AV Club]
And cosy is preferred outside North America:
His office, an extension of his Hertfordshire home, is a cosy mishmash of past and present. [Guardian]
He often cosies up with his grandparents and sings to them when they’re all lounging on one. [Daily Mail]
NAB’s marketing campaign of ”divorcing” the other banks and cosying up to customers will now be put under the spotlight. [Sydney Morning Herald]