Which is the proper spelling of "nonstop?"
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It appears that in the British English corpus the hyphenated version, non-stop music (blue line), is much preferred.
Whereas in American English, the spelling nonstop music (red line) is overwhelmingly preferred, and has been since the 1980s.
Dictionary.com informs that nonstop (without a space) was first used between 1900 and 1905.
Choose whichever spelling you prefer, they are both “correct”.
To dispel doubts on the validity of these Ngram results, which I never claim were 100% accurate, I ran the expression worked nonstop through Ngram Viewer. The blue line represents the American English corpus while the red line is British English, and compared the two dialects together.
There is therefore strong evidence to suggest that American English writers prefer the solid one word (nonstop) compare to its hyphenated version.
The Oxford Guide to Style (2002) does a nice job of identifying where mainstream UK and U.S. style preferences tend to diverge on the issue of how to handle prefixes such as non-:
5.10.2 Prefixes and combining forms
Words with prefixes are often set as one word, but use a hyphen to avoid confusion or mispronunciation, particularly where there is a collision of vowels or consonants:
[Relevant examples:] non-effective non-negotiable
The hyphen is used less in US practice. Words beginning with non- and re-, for example, are often set as one word:
[Relevant examples:] noneffective nonnegotiable
Since nonstop doesn't seem liable to confusion or mispronunciation, and since it doesn't involve a crash of the same consonant (as, for example, non-negotiable does), I imagine that Oxford would have no problem with setting nonstop as one hyphenless word.
The Chicago Manual of Style, fifteenth edition (2003), certainly endorses the closed-up, no-hyphen treatment of most prefix-and-stem combinations, in keeping with the "US practice" that Oxford mentions above. Here is Chicago's advice, in section 7.90, for words containing the non- prefix:
3. Words Formed with Prefixes
Compounds formed with prefixes are normally closed, whether they are nouns, verbs, adjectives, or adverbs.
non: nonviolent, nonevent, nonnegotiable, but non-beer-drinking
Other style guides differ, and of course personal preferences are all over the map; but the coverage in Oxford and Chicago strongly suggests that UK usage is split (or indifferent) as between non-stop and nonstop, while U.S. usage definitely favors the latter.
But just to emphasize an essential point that others have made in comments and answers here, nonstop vs. non-stop vs. non stop is ultimately a style issue, not a matter of correct or incorrect spelling in the sense that, say, clohes vs. clothes is.
Non is a prefix, so using non stop (two words) is incorrect.
Both forms are used, but a Google search of non-stop yields 347,000,000 results, and a Google search on nonstop yields 83,800,000 results. So as far as usage, non-stop is used four times more often than nonstop.
There is no such thing as proper spelling, only what is popular at the moment or idiomatic to a country or region. Consider that the taste of food is spelled either "flavor" or "flavour" and that "tomorrow" used to be spelled "to-morrow" (as was to-day) and I think you'll see that. Personally, I would opt for "non-stop" however I am sure there are many users who will say that is completely and utterly wrong because [fill in the blank with a reason that makes sense.]