Short Answer

Yes, both nose and ear piercings are allowed for women.

What the Qur’an and Sunnah Say

Abdullah ibn Abbas said that the Messenger of Allah, on the day of Eid al-Fitr, went to the women after prayer with Bilal and told them to give charity. They began to throw things [into the sack], each woman throwing in her earrings and necklace.’ (Bukhari #98, #921)

What the Scholars Said

A group of scholars wrote: “There is nothing wrong with women piercing their ears and there is [also] nothing wrong with having the ears of young girls pierced, since this was done in the time of the Messenger of Allah without any rebuke.” (Al-Fatawa al-Hindiyya 5:357)

Imam Ibn Abidin said: “If it [i.e. piercing the nose to wear a nose ring] is considered an adornment for women like in some communities, then its ruling will be similar to the ruling of piercing the ear to wear an earring [i.e. permitted].” (Radd al-Muhtar 6:420)

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Shaykh Mustafa Umar