As per the opinion of Imam A'adham Abu Hanifa (May Allah be well pleased with him), there is no Qunut in Fajr. As per the opinion of Imam Shaf’aee (May Allah shower His Mercy on him), reciting Qunut everyday in the 2nd Raka'at of Salaat Ul Fajr after getting up from the Ruku is a Sunnah, as given in Al Fiqh Alal Madhahib Al 'Arba, Vol. 1, Pg No. 307.
There is a Hadith in Sunan Abu Dawood, which clarifies that the Holy Prophet (Sallallahu alaihi wa sallam) recited Qunut in Fajr for only 1 month and then stopped it. In this way, this commandment of reciting Qunut in Fajr was abrogated.
Translation of Hadith: It has been narrated on the authority of Hadhrat Anas Bin Sireen (May Allah shower His Mercy on him), who narrates on the authority of Hadhrat Anas bin Malik (May Allah be well pleased with him) that the Holy Prophet (Sallallahu alaihi wa sallam) recited Qunut for 1 month and then stopped doing so. (Sunan Abu Dawood, Hadith No: 1233)
Sahih Bukhari has a mention of reciting Qunut in Fajr:
Translation of Hadith: It has been narrated on the authority of Hadhrat Muhammad Bin Sireen (May Allah be well pleased with him), he says that Hadhrat Anas bin Malik (May Allah be well pleased with him) was asked: Did the Holy Prophet (Sallallahu alaihi wa sallam) recite Qunut in Fajr. He said: Yes. (Sahih Bukhari, Vol. 1, Pg No. 136, Hadith No: 1001)
In light of this Hadith, those who follow Imam Shaf’aee (May Allah shower His Mercy on him) recite Qunut in Fajr. In the opinion of the Imams of the Hanafi school of Fiqh, this Hadith of Sahih Bukhari has been abrogated by the previously mentioned Hadith of Sunan Abu Dawood. Now this commandment does not exist.
In light of the Hadith of Sunan Abu Dawood, Allama Badruddin 'Aini Hanafi (May Allah shower His Mercy on him) writes in Umdatul Qari, his famous interpretation of Sahih Bukhari: Earlier there was a commandment of reciting Qunut in Fardh Salaat, later it was abrogated. (Umdatul Qari, Vol. 5, Pg No. 233)
Now about the method of reciting Qunut in Witr. As per the Hanafi school of Fiqh, Qunut is Wajib (compulsory) in Witr. In the 3rd Raka'at, after finishing the recitation of Surah Fatiha and another Surah after it, the Takbir should be said, the hands should be raised to the ears and again bound and then the Qunut should be recited. After this the Ruku should be done. As given in Fatawa A'alamgiri, Kitab Us Salaat, Vol. 1, Pg No. 111).
In the opinion of Imam Shaf’aee, reciting Qunut in the last Raka'at of Witr in the last half of Ramadhan is a Sunnah, as given in Al Fiqh Alal Madhahib Al 'Arba, Vol. 1, Pg No. 307.