There is no authentic Hadeeth which establishes or proves that the Messenger of Allaah (Sallalahu Alayhee Wasallam) raised his hands for the Qunoot of the Witr prayer. The evidence for this is the general Hadeeth of Anas Ibn Maalik RadhiAllaahu Anhu in which he says, “The Messenger of Allaah would only raise his hands for Dua in the prayer for the rain.” (Saheeh an-Nasaa’ee no.1649)
A report from Umar RadhiAllaahu Anhu states that he would raise his hands in the Qunoot (cited by Imaam Bukhaari in Juz Raf ul-Yadain no.97). Yet again there are 2 basic problems. Firstly there is no specification that this hadeeth is related to the Qunoot of Witr And therefore it could very well be pertaining to the Qunoot of the Fajr prayer. Secondly it also contains the weak narrator J’afar Ibn Maimoon who is weak.
However we do find the action of the illustrious companion Abdullaah Ibn Mas’ood raising his hands as Imaam Bukhaari has cited in his Juzz Raf ul-Yadain (no.99) as cited by Imaam Abdur Rahmaan Mubaarakpooree in his monumental explanation of Tirmidhee (refer to his Tuhfatul Ahwadhee 2/580) yet this narration is weak due to problems contained in it’s chain due to Laith Ibn Abee Sulaim whose memory deteriorated and he was also accused of being a mudallis narrator.
Aamir bin Shibal al-Jarmee narrates that he saw Abu Qilaabah raise his hands for the Qunoot.(Abu Qilaabah was a trustworthy successor) (Sunan al-Kubraa 3/41)
Abu Haatim ar-Raazee (d.277H) said, “Abu Zur’ah ar-Raazee (d.264H) asked me, “Do you raise your hands for Qunoot (of witr)? I replied, “No” and then I asked him do you raise your hands and he replied yes. So I asked him what is your evidence?
Abu Zur’ah said: the hadeeth of ibn Mas’ood
Abu Haatim said: Laith ibn Abee Sulaim narrated it (ie a weak narrator)
Abu Zur’ah said: the hadeeth of Abu Hurairah
Abu Haatim said: Ibn Laheeya’ah narrated it (ie again a weak narrator)
Abu Zur’ah said: the hadeeth if Ibn Abbaas
Abu Haatim said: A’uf has narrated it (again a weak narrator)
Then Abu Zur’ah asked me. “what is your evidence for not raising your hands?”
Abu Haatim said: the hadeeth of Anas that verily the Messenger of Allaah (Sallalahu Alayhee Wasallam) would never raise his hands for any dua except in the rain prayer , and then he (Abu Zur’ah) remained silent.
(Taareekh Baghdaad (2/76 no.455) and also cited by Imaam Dhahabee in Siyaar al-A’laan an-Nabula 13/253)
The Imaam of Ahlus Sunnah Ahmad ibn Hanbal and Imaam Ishaaq ibn Rahawaihah opined in the raising of the hands for the Qunoot (Masaa’il Abee Dawood pg.66, Masaa’il Ahmad Waa Ishaaq 1/211 no.465 and 2/59 no.3468 narrated via Ishaaq ibn Mansoor al-Kausaj also Masaa’il pg.23 of Marwazee as cited by Allaamah al-Albaanee)
Imaam al-Albaanee has cited the opinion of Imaam Ahmad and Ishaaq ibn Rahawaihah with reference to the raising the hands for the Qunoot an-Nawaazil as their is authentic evidence for this as can be read from his Sifah Salah.
So it seems like some scholars raised their hands and other did not, based on their understanding of the texts but the opinion with precedence is not raising the hands due to the hadeeth of Anas ibn Maalik RadhiAllaahu Anhu and other texts.
The people who raise their hands in the Qunoot of Witr and supplicate, do so based on analogical reasoning on the Qunoot an-Nawaazil (in which raising the hands is established)
There is no direct clear authentic evidence which mentions the Prophet (Sallalahu Alayhee Wasallam) raised his hands for the Qunoot of witr but if someone wishes to do so based on analogical reasoning then this is also permissible as the Scholars have said.
Raising the hands for the Qunoot of calamities is established from the Sunnah. (Musnad Ahmad (3/137 no.12429)