There is no such word as “theirselves” (and you certainly can’t spell it “theirselfs” or “thierselves”); it’s “themselves.”
In traditional English there is no correct singular form of this non-word; instead of “theirself” most people use “himself” or ”herself.” If you encounter “themself,” it’s probably not a mistake but a deliberate choice to avoid specifying gender, following the pattern of using singular “they” for the same reason.
BUY THE BOOK!