What is the ruling on swearing by other than Allah, such as "by your conscience," or "by your prayer," or "you are in a difficult situation if you do such and such," which is common among women and children?

It is not permissible to swear by other than Allah, whether by prayer (salat), by one's 'dhimma' (responsibility/conscience), by 'haraj' (distress/difficulty), or by other creatures. Swearing must be by Allah alone or by His attributes, for the Prophet, peace and blessings be upon him, said: "Whoever takes an oath should swear by Allah or remain silent," and "Whoever swears by something other than Allah has committed shirk (associating partners with Allah)," and "Whoever swears by other than Allah has disbelieved or committed shirk," and "Whoever swears by the trust (amanah) is not one of us." However, if one says "fi dhimati" (on my conscience/responsibility) as an indication of trust, that is not an oath. It is incumbent upon the Muslim to guard his oaths and not to swear by other than Allah except when necessary.