Qunoot in the Witr prayer is established before the rukoo from the Messenger of Allaah (Sallalahu Alayhee Wasallam) and this is what most of the narrations indicate aswell as some reports mentioning it after the rukoo.
It is narrated from Ubayy Ibn Ka’ab RadhiAllaahu Anhu who said, “Verily when the Messenger of Allaah (Sallalahu Alayhee Wasallam) prayed the Witr, he would recite the Qunoot before the Rukoo.” (Ibn Maajah (no.1182), Sunan an-Nasaa’ee (3/235 no.1700), Sunan Daarqutnee (2/31 no.1644), Dhiya Maqdisee has also transmitted this in al-Mukhtarah Imaam al-Albaanee authenticated it refer to Saheeh Ibn Maajah (no.970) and Irwaa ul-Ghaleel (no.462)
Hasan RadhiAllaahu Anhu narrates, “The Messenger of Allaah (Sallalahu Alayhee Wasallam) taught me to recite the following after completing my recitation in the witr prayer: Allaahuma Mahdinee….” (Kitaab at-Tawheed (2/91) of Ibn Mandah and Irwaa ul-Ghaleel (2/168) (meaning the supplication which is the Qunoot is to recited before the rukoo)
Alqamah narrates “Abdullaah ibn Masood RadhiAllaahu Anhu and the companions of the Messenger of Allaah (Sallalahu Alayhee Wasallam) would recite the Qunoot in the witr before the rukoo.” (Musannaf ibn Abee Shaybah (2/306), The late Albanian hadeeth master, the Allaamah Muhammad Naasir ud deen authenticated this narration and said it chain was Saheeh according to the conditions of Imaam Muslim see Irwaa ul-Ghaleel (2/166). Haafidh Ibn Hajr also authenticated it and said it is Hasan in ad-Diraayah (1/94)
Aasim Ahwal said, “I asked Anas bin Malik about the Qunoot. Anas replied, “Definitely it was (recited)”. I asked, “Before bowing or after it?” Anas replied, “Before bowing.” I added, “So and so has told me that you had informed him that it had been after bowing.” Anas said, “He told an untruth (i.e. “was mistaken,” according to the Hijazi dialect). Allah’s Apostle recited Qunut after bowing for a period of one month.” Anas added, “The Prophet sent about seventy men (who knew the Quran by heart) towards the pagans (of Najd) who were less than they in number and there was a peace treaty between them and Allah’s Apostles (but the Pagans broke the treaty and killed the seventy men). So Allah’s Apostle recited Qunut for a period of one month asking Allah to punish them.” (Bukhaari, the book of war expedition (no.4096), Abu Awaanah (2/285), Daarimee (1/374), Sharh Ma’anee al-Aathaar of Tahaawee (1/143), Sunan al-Kubraa (2/207) of Baihaqee and Musnad Ahmad (3/167)
The narration above proves the Qunoot of Nawaazil ie calamity or wishing good for the Muslim Ummah or cursing the enemies of Islaam ie the mushriks and kaafirs then this is something that is done after the rukoo which is commonly known as Qunoot an-Naazilah. As for the Qunoot before the rukoo then that is more specific to the Qunoot for Witr.
There is however a narration from Anas Ibn Maalik RadhiAllaahu Anhu in which he mentions the saying of the Qunoot before or after the rukoo, then this is pertaining to and is specific to the Qunoot for the Fajr prayer and not for the Witr prayer as is specified in the Hadeeth itself.(Ibn Maajah (no.1184), Imaam al-Albaanee authenticated it refer to Saheeh Ibn Maajah (no.971) and Irwaa ul-Ghaleel (2/160)
There are some reports from Umar RadhiAllaahu Anhu that he would recite the Qunoot after the Rukoo and he would raise his hands to do so. (cited by Imaam Bukhaari in his Juzz Raf ul-Yadain (no.97).
However firstly this narration is weak due to the Narrator Ja’afar Ibn Maimoon who is considered to be weak according to the majority of the scholars of Hadeeth. Secondly according to the categorisation of this Hadeeth by Imaam Ibn Abee Shaybah he understood this narration pertaining to the Qunoot of the Fajr prayer and hence has therefore categorised this Hadeeth to be placed under the chapter heading The Qunoot for the Fajr. (refer to Imaam Ibn Abee Shaybahs Musannaf (2/316 no.7040)
There is another report from Abdullaah Ibn Masood RadhiAllaahu Anhu via Laith ibn Abee Sulaim that Ibn Masood RadhiAllaahu Anhu would raise his hands before the rukoo and recite the Qunoot. (Juzz Raf ul-Yadain pg.173-173 no.99) but this is weak due to Laith ibn Abee Sulaim
There is a narration from Hasan RadhiAllaahu Anhu in Mustadrak al-Haakim (3/172) and Imaam Baihaqees Sunan al-Kubraa (3/38-39) which establishes the Qunoot after the rukoo however this is weak according to some scholars of hadeeth whereas others like the late Albanian Hadeeth Master Muhammad Naasir ud deen authenticated it and graded it Hasan. This may be understood due to the text of the aforementioned hadeeth as the hadeeth has been narrated via other chains and hence him grading it Hasan.
Imaam Tirmidhee said the people of knowledge have differed with regards to the Qunoot of the Witr Abdullaah ibn Masood RadhiAllaahu Anhu opined it should be read before the rukoo and this is position of some of the people of knowledge like Sufyaan ath-Thawree, Ibn al-Mubaarak and Ishaaq (ibn Rahawaihah) and the people of Koofah. (Sunan Tirmidhee, Chapter what is said with regards to the Qunoot for the Witr (1/106)
Shaykh al-Islam Ibn Taymiyyah said, “With regard to Qunoot: there are two extreme views and one middle (or moderate) view. Some say that Qunoot should only be recited before bowing and some say that it should only be recited after bowing. The fuqaha’ among the scholars of hadeeth, such as Ahmad and others, say that both are allowed, because both are mentioned in the saheeh Sunnah, but they preferred reciting Qunoot after bowing because this is mentioned more often.”
(Majmoo’ al-Fataawa (23/100)
Allaamah Imaam al-Albaanee has also opined that the Qunoot should be recited before the rukoo (refer to his Sifah Salaah, Shaikh al-Albaanee goes onto cite further references from the various books of hadeeth)
Shaykh Ibn ‘Uthaymeen said “According to most of the ahaadeeth and according to the opinion of most of the scholars, Qunoot should come after the rukoo’, but if a person recites it before the rukoo’ that is o.k. He has the choice of bowing when he has finished reciting from Qur’aan, and when he stands up from bowing and says “Rabbana wa laka’l hamd,” then he may recite Qunoot, as described in most of the reports from the Prophet (peace and blessings of Allaah be upon him) and as stated by most of the scholars; or he may recite Qunoot when he has finished reciting Qur’aan, then say Allaahu akbar and bow. Both ways were narrated in the Sunnah.
(ash-Sharh al-Mumti’ by Ibn ‘Uthaymeen, 4/65)