In Ṣalāh the hands should be raised up to the level of the shoulders: 

1. on saying the takbeer for beginning the Ṣalāh,

2. on saying the takbeer for bowing (rukoo')

3. and on saying "sami allāhu liman hamidah (rising from rukoo')

4. where there are two tashahhud sittings in the salāh (i.e. a Ṣalāh of more than 2 rak'āt), then the hands should be raised on raising up from the first sitting. 

The text that follows gives evidence of the above from the authentic Sunnah of the Prophet (ﷺ). 

Saheeh Al-Bukhārī

Hadeeth no.735. Narrated Salim bin Abdullāh:

"My father said, 'Allāh's Apostle ﷺ used to raise both his hands up to the level of his shoulders when opening the prayer; and on saying the Takbeer for bowing. And on raising his head from bowing he used to do the same and then say 'Sami Allāhu liman hamida, Rabbana walaka-l-hamd.' And he did not do that (i.e. raising his hands) in prostrations.'" 

Hadeeth no.736. Narrated Abdullāh bin Umar:

"I saw that whenever Allāh's Apostle ﷺ stood for the prayer, he used to raise both his hands up to the shoulders, and used to do the same on saying the Takbeer for bowing and on raising his head from it and used to say, 'Sami Allāhu liman hamidah'. But he did not do that (i.e. raising his hands) in prostrations."

Hadeeth no.737. Narrated Abu Qilaba:

"I saw Malik bin Huwairith saying Takbir and raising both his hands (on starting the prayers and raising his hands on bowing and also on raising his head after bowing. Malik bin Huwairith said, 'Allah's Messenger ﷺ did the same.'"

Hadeeth no.738. Narrated Abdullāh bin Umar:

"I saw Allah's Messenger ﷺ opening the prayer with the Takbir and raising his hands to the level of his shoulders at the time of saying the Takbir, and on saying the Takbir for bowing he did the same; and when he said, 'Sami`a l-lahu liman hamidah', he did the same and then said, 'Rabbana wa laka lhamd.' But he did not do the same on prostrating and on lifting the head from it.'"

Hadeeth no.739. Narrated Nafi:

"Whenever Ibn 'Umar started the prayer with Takbeer, he used to raise his hands: whenever he bowed, he used to raise his hands (before bowing) and also used to raise his hands on saying, 'Sami Allāhu Liman hamidah', and he used to do the same on rising from the second Rak'ah (for the 3rd Rak'ah). Ibn 'Umar said: 'The Prophet ﷺ used to do the same.'"

Saheeh Muslim

Hadeeth no. 401.  Narrated Wa'il ibn Hujr:

"Wā'il saw the Apostle of Allāh ﷺ raising his hands at the time of beginning the prayer and reciting takbir, and according to Hammam (the narrator), the hands were lifted opposite to ears. He (the Prophet) then wrapped his hands in his cloth and placed his right hand over his left hand. And when he was about to bow down, he brought out his hands from the cloth, and then lifted them, and then recited takbeer and bowed down, and when (he came back to the erect position) he recited: 'Allāh listened to him who praised Him.' And when prostrated, he prostrated between the two palms."  

Hadeeth no. 391b.  Malik b. Huwairith reported:

"The Messenger of Allah ﷺ raised his hands apposite his ears at the time of reciting the takbir (i. e. at the time of beginning the prayer) and then again raised his hands apposite the ears at the time of bowing and when he lifted his head after bowing he said: 'Allāh listened to him who praised Him', and did like it (raised his hands up to the ears)." (See also 390c, transmitted by Qatada)

Hadeeth no. 391a:

Abu Qilaba reported that he saw Malik b. Huwairith raising his hands at the beginning of prayer and raising his hands before kneeling down, and raising his hands after lifting his head from the state of kneeling, and he narrated that the Messenger of Allāh ﷺ used to do like this.

Hadeeth no. 390 b. Ibn Umar reported:

"The Messenger of Allah ﷺ, when he stood up for prayer, used to raise his hands apposite the shoulders and then recited takbir (Allah-o-Akbar), and when he was about to bow he again did like it and when he raised himself from the rukoo' (bowing posture) he again did like it, but he did not do it at the time of raising his head from prostration."

Hadeeth no. 390 a. Salim narrated it on the authority of his father who reported:

"I saw the Messenger of Allāh ﷺ raising his hands apposite the shoulders at the time of beginning the prayer and before bowing down and after coming back to the position after bowing. but he did not raise them between two prostrations."

Fat`h-ul-Barī (Commentary to Saheeh Al-Bukhārī)

Under the chapter heading “raising of hands at the takbeer, and at rukoo` & at rising from it” Al-Hāfidh ibn Hajr wrote:

“Al-Bukhārī mentioned that the raising of hands at rukoo‘ and at rising from rukoo‘ is reported from 17 Companions. Al-Hākim and Abu al-Qāsim ibn Mandah mention that from amongst the ones who narrate it are the 10 promised paradise. Our Shaykh, Abu al-Fadl al-Hāfidh [al-I‘rāqī], mentioned that he investigated all those who reported it from amongst the Companions and they reached 50 [Companions].” (2/280)

Sifāt Salātun Nabī 

Under the chapter heading "The Rukoo' (Bowing)", Imām Muhammad Nāsiruddin (rahimahullāh) presented the following description from the sunnah:

"After completing his recitation, he ﷺ would pause for a moment, then raise his hands, say takbeer, and make rukoo'."

This raising of the hands is the madhhab of the three Imāms Mālik, Shāfi`i and Ahmad, and of the majority of scholars of Hadeeth and Fiqh.  Imām Mālik (rahimahullāh) practiced it right up to his death, as reported by Ibn `Asākir (15/78/2).  Some of the Hanafis chose to do it, among them `Isām bin Yusuf Abu `Asamah al-Balkhi (d. 210), a student of Imām Abu Yusuf (rahimahullāh). `Abdullāh bin Ahmad reported from his father in his Masā'il (p. 60), "It is related from `Uqbah bin `Aamir that he said about a man raising his hands during prayer, 'He earns ten good deeds for each such movement'." This is supported by the Hadeeth qudsi, "... he who intends a good deed and then does it, Allāh writes it down with Himself as from ten to seven hundred good deeds", transmitted by Al-Bukhāri & Muslim. See Saheeh at-Targheeb, no. 16.