I’ve been reading draft student papers. One extremely common locution goes like this: “If a person does [something] then their liability will be [whatever].” That “their” is there because students don’t want to say “his” or “her” nor use the clunky “his or her”. English doesn’t currently offer a neuter word; “their” is a plural when the grammar requires a singular, but to my students’ ear that is less of an issue than picking a single gender to refer to both.
Why exactly they don’t pluralize the whole thing (“If people do [whatever] then their liability will be [whatever]”) I don’t know.
This language shift suggests that at some time in the future the non-prescriptivist definition of “their” will shift to include a role a neuter singular possessive. But I don’t believe we are there yet on “their” so I’m marking “their” up whenever I find it misused.
Then again, I may be behind the times: Dictionary.com already offers “their” a secondary singular meaning:
2. (used after an indefinite singular antecedent in place of the definite masculine form his or the definite feminine form her): Someone left their book on the table. Did everyone bring their lunch?