Anybody, anyone, somebody, someone, nobody, no-one, everybody and everyone are used with singular verbs.
Is everybody ready?
(NOT Are everybody ready?)
However, we often use they, them and their to refer to these words, especially in an informal style.
If anybody calls, tell them I'm out, but take their name and address. Nobody phoned, did they?
Somebody left their umbrella behind yesterday. Would they please collect it from the office?
Everybody thinks they're different from everybody else.
They, them and their are not plural in sentences like these. They mean 'he or she', 'him or her' and 'his or her'. In a more formal style, we usually use he, him and his (meaning 'he or she', etc).
When somebody does not want to live, he can be very difficult to help.