I use 'they' as a third person singular in a number of situations. I know it is grammatically incorrect, but in a situation where a real or hypothetical group of people contains men and women, and you want to talk about one identified person out of that group, it is sometimes better than 'he or she', I think.
Eg if I head up a team which contains both men and women. I am talking to someone about how my team works. I would say 'I will give this to one of my team and ask them to check it'. If I say 'him' then it assumes I have chosen a male person (when in reality I haven't yet chosen anyone) or it assumes all my team is male (which isn't true). If I say 'ask him or her to check it' it sounds clumsy and makes it obvious I don't have anyone in mind.
What do other people do?
(Question prompted by the pronouns thread where someone said they [she?] disliked using the they as a singular).