Chicago Manual of Style, 15th ed., sect. 7.90.3, pp. 306ff.:
Compounds formed with prefixes are normally closed, whether they are nouns, verbs, adjectives, or adverbs.
A hyphen should appear, however, (1) before a capitalized word or a numeral, such as sub-Saharan, pre-1950; (2) before a compound term, such as non-self-sustaining, pre–Vietnam War (before an open compound, an en dash is used; see 7.83); (3) to separate two i
's, two a
's, and other combinations of letters that might cause misreading, such as anti-intellectual, extra-alkaline, pro-life; (4) to separate the repeated terms in a double prefix, such as sub-subentry; (5) when a prefix or combining form stands alone, such as over- and underused, macro- and microeconomics.
non: nonviolent, nonevent, nonnegotiable, but
Garner, Modern Amer. Usage, p. 540:
J. Hyphen [-].
Generally, AmE is much less hospitable to hyphens than BrE. Words with prefixes are generally made solid: displeasure (not dis-pleasure), preshrunk (not pre-shrunk), postdebate (not post-debate), preordain (not pre-ordain). This no-hyphen style seems aesthetically superior, but reasonable people will differ on such a question. They can agree, however, that the hyphen must appear when an ambiguity or miscue is possible without it—e.g., pre-judicial (career), re-sign (the letter). See CO- & RE- PAIRS.
Fowler/Burchfield, Modern English Usage, 3rd ed., p. 525:
non- ... In AmE such compounds are likely to be written solid, i.e. without a hyphen.
I hate to spoil this happy consensus, but sorry, one more dissenting voice.
Obviously it's totally wrong to write 'non' as a separate word. Another similar common mistake is 'mid' alone, which should also always be a prefix, not a word.
However, writing 'non-' with a hyphen doesn't look much better to me, at least in AE. I've never heard of that supposed rule about adding a hyphen before a vowel, nor do I find any such rule in the standard reference sources. (The one about the capital letter does make sense, and Chicago confirms it.)
Really, for spelling, the best source is often just a dictionary. Check 'noninvasive' or whatever in M-W or AHD and see if it isn't, er, nonhyphenated. (-;http://www.m-w.com/dictionary/noninvasive
And in fact, even your computer spell-check should be able to handle that one, I would think, though I wouldn't be absolutely surprised if its word lists didn't cover every conceivable combination with non-.
Selkie, all this is assuming you're writing AE, that is. Anyone writing BE would need to check British dictionaries or style guides. In general, BE has traditionally used more hyphens with prefixes than AE, and non- seems to be one of the prefixes that at least used to be hyphenated in BE. But on the other hand, AE is fast influencing BE especially in areas such as hyphenation and word division, perhaps because the AE rules tend to be simpler and more consistent. So it's probably a good idea to see how the Guardian or the Times or something currently handles a case like this, as preferences may have changed even in the last 5 or 10 years.
Morning, ale. I'm with the dictionary on 'hotheaded,' BTW -- not sure what Paul Brian was thinking there. (-: