A pronoun should agree in number and gender with its antecedent noun. Note that “their,” “them,” and “they” are (for the moment, at least, and by conservative standards) plural pronouns; try not use these pronouns if the preceding noun is singular. “They” is widely used as a singular pronoun in spoken English and is not uncommon in written English. Some scholars do consciously use “they” or “hu” as a third-person singular pronoun. It’s clear that English needs a gender-neutral singular pronoun. I expect that standard usage will change in the next decade. In March 2015, the Swedish Academy announced the addition of the gender-neutral pronoun “hen” to the Svenska Akademiens ordlista (SAOL), the official dictionary of the Swedish language.
In the meantime:
- Avoid: A modern actor usually strives to find psychological validity in their Shakespearean character.
- Grammatically correct but clunky and possibly offensive: A modern actor usually strives to find psychological validity in his [or “her,” “the,” “a,” or “Shakespeare’s”] character.
- Right (and less clunky and unlikely to offend): Modern actors usually strive to find psychological validity in their characters.
Note that “each” and “neither” take a singular verb in Canadian usage.
- Each of the students in the class gives two presentations.
- Neither of the students was present on Monday.