What is the ruling on the oaths of this woman who swears by her children, then forgets some of her oaths or retracts them, and is an expiation (kaffarah) obligatory upon her in every instance?

Frequent swearing is forbidden by Islamic law, even if the swearer is truthful, because it leads to trivializing Allah Almighty. Whoever swears frequently is a "Halaaf Maheen" (one who swears often and is contemptible). There is a difference between "Yamin al-Laghw" (oath of idle talk), which incurs no expiation and is what slips from the tongue unintentionally, and "Yamin al-Mun'aqidah" (deliberate oath), which necessitates expiation upon breaking it. If the breaking of the oath was due to forgetfulness, ignorance, or coercion from the party for whom the oath was sworn, then no expiation is required from the swearer.