What is the ruling on one who swore falsely upon the Mus'haf out of fear of divorce, and is expiation for an oath (kaffarat yamin) incumbent upon her?

As long as the oath is intended to preserve the marital relationship, there is no harm in it. However, it is better to resort to equivocation (tawriya) to avoid a false oath. A false oath to avert significant harm is permissible for an overriding benefit.