What is the evidence from the Qur'an and Sunnah (whatever is reported from the Prophet) on the permissibility of divorce being in the hands of the wife?
The origin in this issue is that divorce is in the hands of the husband, for Allah (Exalted be He) says:
O Prophet (صلى الله عليه وسلم)! When you divorce women, divorce them at their ‘Iddah (prescribed periods).
However, if the husband (Part No. 20; Page No. 12) delegates his right of divorce to his wife, and she divorces herself afterwards, in this case the divorce will occur. May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.