You need to check if this is actually a "legal separation".  From what I understand, those are extremely rare.

From what I understand, those are basically a divorce, but for some reason the taxpayer's don't want to use the term "divorce" (perhaps religious reasons).

If it actually is a legal separation, then it is treated just like a divorce.  So the taxpayer's would be Single (or Head of Household, if they qualify).  But again, actual legal separation is quite rare.