It is narrated from Abdullah bin Umar (may Allah be pleased with him) that the 'iddah of a woman who obtains khul' is like the 'iddah of a divorced woman.
Imam al-Bayhaqi (may Allah have mercy on him) said: This is the view of Sa'id bin al-Musayyab, Sulayman bin Yasar, al-Zuhri, al-Sha'bi and the majority (al-jama'ah).
Hadith Referenceالسنن الكبرى للبيهقي / كتاب العدد / 15596
It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with him) that the wife of Thabit bin Qays (may Allah be pleased with him) obtained khul' from him, and the Prophet (peace and blessings be upon him) set "her 'iddah at one menstrual period."
Likewise Ali bin Bahr, Isma'il bin Yazid al-Basri and others narrated it from Hisham, from Ma'mar, as connected (mawsul). [Da'if (weak)]
Hadith Referenceالسنن الكبرى للبيهقي / كتاب العدد / 15597
It is narrated from Ikrimah (may Allah have mercy on him) that the wife of Sayyiduna Thabit bin Qays (may Allah be pleased with him) obtained khul' from him, and the Prophet (peace and blessings be upon him) fixed "her 'iddah at one menstrual period." This has also been narrated through two other weak chains, and it is not permissible to use the likes of them as evidence.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب العدد / 15598
It is narrated from Sayyidah Rubayyi' bint Mu'awwidh (may Allah be pleased with her) that she obtained khul' in the blessed time of the Prophet (peace and blessings be upon him), and the Prophet (peace and blessings be upon him) ordered her, or she was ordered, "to observe an 'iddah of one menstrual period."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب العدد / 15599
It is narrated from Rubayyi' bint Mu'awwidh bin Afra' (may Allah be pleased with her) that she sought khul' (separation) from her husband, and she was ordered to observe a waiting period of one menstrual cycle.
This is more correct, and it does not mention who ordered her, nor the words "in the time of the Prophet (peace and blessings be upon him)." We have narrated in the Book of Khul' that she sought khul' from her husband in the time of Uthman bin Affan (may Allah be pleased with him).
Hadith Referenceالسنن الكبرى للبيهقي / كتاب العدد / 15600
It is narrated from Nafi' that Ibn Umar (may Allah be pleased with them both) informed him that Rubayyi' bint Mu'awwidh bin Afra' (may Allah be pleased with her) took khul' from her husband in the time of Uthman (may Allah be pleased with him). Her paternal uncle, Mu'adh bin Afra' (may Allah be pleased with him), went to Uthman (may Allah be pleased with him) and said: The daughter of Mu'awwidh has taken khul' from her husband today; may she move out? Uthman (may Allah be pleased with him) said: She may move out, and there is no waiting period ('iddah) upon her, but she must not marry until she has passed one menstrual cycle. Abdullah bin Umar (may Allah be pleased with them both) said: Uthman (may Allah be pleased with him) is the eldest of us and the most knowledgeable of us. This narration makes clear that it was Uthman (may Allah be pleased with him) who ordered her to do that, while the apparent meaning of the Book concerning the waiting period of divorced women covers the woman who takes khul' and others, so it is more appropriate; success is from Allah, and Allah knows best.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب العدد / 15601