"Everyone . . . their"
In general, pronouns should agree in number and gender with their antecedents. Speakers and writers of English most frequently depart from this rule when they sense that it would be inappropriate to limit the person picked out by the antecedent to a single gender. The problem here has long been recognized; the original edition of the Oxford English Dictionary noted matter-of-factly that the "pronoun referring to every one is often pl[ural]: the absence of a sing[ular] pron[oun] of common gender rendering this violation of grammatical concord sometimes necessary," and gave, among its examples, this sentence from 1877:
Under "Every body," the OED quoted Lord Byron (1820): "Every body does and says what they please."
The Guide treats the phenomenon of "singular their" on the Conventions page. Despite the widespread use and evident convenience of the "singular their" construction, it is not widely accepted usage in writing, particularly formal writing, and we recommend that you avoid it, preferring instead one of several other easily exercised workarounds for the lack, in English, of a neuter singular third-person pronoun. Take the following sentence:
The plural pronoun their does not accord in number with its singular antecedent noun, student. Not so many years ago, students would have been advised to substitute his for their, achieving numerical concord at the expense of truth (except, of course, at all-male colleges).
A better solution, however, is to change what a student needs to what students need. The reason that students fills in readily for a student is that a student in this sentence doesn't name a particular individual but a hypothetical individual - which is to say, in effect, a class of individuals. When a noun does name a particular person, it makes perfect sense to use his or her as required by the person's gender. Indeed, in this case using the correct singular pronoun can eliminate ambiguity. For example:
Just whose secret was revealed here? It might have belonged to the student, to a person of unknown gender, or to a group of people. Perhaps the context of the sentence would have made the owner's identity clear - but perhaps not.
The uncertainty of some writers with respect to pronoun agreement leads to the worst of all faults: rapid pronoun shifting. For example:
If you start out with one, stick with it all the way through the sentence:
If the many repetitions of one bother you, you might want to try a different approach to the sentence altogether. In speech and informal writing, a common approach to this kind of sentence is to use the pronoun you throughout. Thus:
To Americans, this sentence is likely to seem more natural than the sentence that repeats one. However, some professors regard this use of you as unsuited to formal writing, preferring to see you reserved for those cases where the writer means to refer to a specific individual, such as the reader. For more on this issue, see the Formal and Informal Writing page.
The relative pronouns who, which, and that create some interesting usage quandaries. On the Myths page, The Guide deals at length with the choice between that and which when introducing restrictive and nonrestrictive clauses. Here we note merely that you should reserve that and which for objects, employing who and whom for persons. This is a fairly recent usage preference - consider the epigraph from Samuel Johnson on The Guide's Essay Exams page and Mark Twain's story "The Man That Corrupted Hadleyburg" - but many of your professors will nonetheless expect you to adhere to it.