A parallel grammatical convention to subject/verb agreement is pronoun/antecedent agreement—requiring the writer to use pronouns that agree in number with the antecedent (the noun to which the pronoun refers) of that pronoun. [EDIT]
Writers must be careful with keeping pronouns and their antecedents parallel in number. Singular nouns require a singular pronoun, and both must be plural, of course.
A common problem in usage for many is the fear of assigning a gender to a nonspecific noun that is singular. See the example below:
Nonstandard: A student left their book in the gym. (Non-specific gender of “student” leads to “their,” which is nonstandard since it is plural, but it seems to agree with “student” since it doesn’t specify gender.)
Conventions require that writer agree pronouns and antecedents by number. If a writer is being hypothetical, prefer plural nouns and pronouns to avoid number errors and to avoid gender issues in language usage.
Pronoun/antecedent agreement is the focus of a debate about language conventions and sexist language use. The standard convention for using a pronoun to refer to a singular noun has been the third person singular pronouns “he,” “him,” and “his,” resulting in sexist language. Many have advocated awkward constructions such as “he/she,” “him/her,” and “his/hers.” But many balk at the awkward forms. Each writer must make a decision about these usage questions, paying close attention to the guidelines concerning the use of sexist language in course or when submitting work for publication.
Again, be cautious with indefinite pronouns:
Always singular: each, either, neither, one, everyone, no one, nobody, anyone, anybody, someone, somebody, everybody, much
Always plural: several, few, both, many, others.
Either singular or plural, depending on reference: some, any, none, all, most.
On-line help with pronoun/antecedent agreement is available here: