Is a woman sinful if she asks for a separation from her husband who constantly threatens her with divorce at the slightest problem, and what is the ruling of Islamic law on a husband who does that?

Spouses must live together in kindness. A husband threatening his wife with divorce is a blameworthy act. If the husband's repeated threats of divorce cause psychological harm to the wife, then she bears no sin for seeking a divorce or khulʿ. The warning in the hadith (definition), "Any woman who asks her husband for divorce without any [valid] reason, the scent of Paradise will be forbidden to her," does not apply to her, because she has a valid excuse. Although divorce is disliked in Islam, it is permissible when reconciliation becomes impossible. It is preferable to be patient and to try to reach an understanding with the husband, explaining the psychological impact of his threats, for marital life is sustained through patience and overlooking faults.