The indefinite pronouns no one and nobody are largely interchangeable. Garner (1998) notes that no one is more formal and literary, a judgement supported by this corpus analysis. Both terms, however, are apt to appear without controversy in almost any kind of writing.
No one, meaning no person, is spelt with two words. The hyphenated no-one is a common variant, especially in informal contexts, though it is less to my taste than the traditional two-worded form. The diaeretic noöne is unlikely to enter common usage. The practice of writing no one as noone may have resulted from its virtual synonymity with the one-worded nobody; from its connection to the similarly unified everyone, anyone and someone; or from the tendency for the morphology of many compound words to go from A B to A-B to AB.
Noone is a decidedly strange spelling of no one. To my eyes, today, it is wrong, but no one can say for sure what usage will be accepted in 50 years’ time. Noone implies the monosyllabic pronunciation /nuːn/, especially to non-native speakers of English. (Mind you, I have yet to hear anyone mispronounce cooperate.) Searches for “noone” on Bartleby.com turned up a small number of results, all of them the archaic spelling of noon.
Moreover, noone immediately suggests some specific person called Noone, e.g. the actor Nora-Jane Noone or the musician Peter Noone. Thus it may lead to momentary ambiguity or to additional meanings that are both unintended and comic:
Noone loves me, but I have my eye on Sullivan.
Noone saw Noone leave the room.
Noone was behind the tree, so I discreetly relieved myself before rejoining the others.
You see the problem.
Now, a few notes on usage.
Indefinite pronouns (no one, everyone, anybody, etc.) usually take singular verbs but can be referred to by singular or plural pronouns (they, them, their). If you follow an indefinite pronoun with a plural pronoun, you scupper notional agreement (AKA “concord”), but you avoid awkward constructions such as s/he and his or her, as well as the accusations of sexism habitually slung at the notoriously gender-specific he, his and him.
Sometimes the singular form will be called for, and it is preferred by some writers, but there is nothing grammatically wrong with the plural.
“Nobody remembers a journalist for their writing” – Richard F Shepard
“[N]o one can ever be in love more than once in their life” – Jane Austen, in Sense and Sensibility
“Nobody here seems to look into an Author, ancient or modern, if they can avoid it” – Lord Byron, in a letter
This last quote is cited in The Merriam-Webster Dictionary of English Usage, which adds that Byron’s “Nobody here” could only have meant males. Yet he opted for genderless they, and it seems altogether natural and sensible. Elsewhere, MWDEU states that “the plural they, their, them with an indefinite pronoun as referent is in common standard use”. Writing about any, anyone and anybody, Robert Burchfield points out that “popular usage and historical precedent favour the use of a plural pronoun”. In adopting the singular use of “plural” they, Byron is in good company.
So, would you write “No one in their right mind”, “No one in his right mind”, “No one in her right mind”, “No one in his or her right mind”, “No one in zer right mind”, or what? My advice is to approach these options with an open mind; to be aware of, but not cowed by, those who decry singular-they constructions; and to let context, meaning and good sense guide your decision.