Allama Ibn Rushd Maliki (May Allah shower His mercy on him) writes in Bidayatul Mujtahid:
Translation: According to a report Imam Malik, Imam Abu Hanifa, Imam Shafa’ee and Imam Ahmed bin Hambal (May Allah be well pleased with him) have agreed that Taraweeh is of 20 Raka’at apart from Witr.
(Bidayatul Mujtahid, Vol 1, Pg No: 210)
20 Raka’at Taraweeh in Shafa‘ee school of Fiqh
First proof
Allama Ibn e Hajr Haithami Makki Shafa’ee (May Allah shower His mercy on him) writes in Al Minhaj Ul Qaweem:
Translation: Nevertheless, the Sahabah (May Allah be well pleased with them) have concurred and agreed on 20 Raka’at of Taraweeh.
(Al Minhaj Ul Qaweem, Pg No: 138)
Second proof
Allama Abul Walid Sulayman bin Khalaf Andusi Malki (May Allah shower His mercy on him) writes in his book Al Muntaqa Sharh Mawatta:
Translation: Based on the Hadith of Hadhrat Yazid bin Ruman (May Allah be well pleased with him): Imam Shafa’ee (May Allah shower His mercy on him) has ruled that Taraweeh is of 20 Raka’at apart from Witr.
20 Raka’at of Taraweeh in the Hambali school of Fiqh
Allama Ibn Qudama Hambali (May Allah shower His mercy on him) writes in his book Al Kafi:
Translation: It is a Sunnah that Taraweeh should be offered as congregational prayer of 20 Raka’at and the Imam should lead the people in Witr also.
(Al Kafi, Kitab Us Salaat)
36 Raka’at – A distinction of the people of Madina
Translation: The Shafa‘ee Fuqaha say that the people of Madina offered 36 Raka’at of Taraweeh. This is because 20 Raka’at of Taraweeh has 5 breaks. The people of Makka perform 1 round of Tawaaf after each break. The people of Madina started offering another 4 Raka’at in each break to become equal to the people of Makka in virtuous acts. This is not allowed for anybody apart from the people of Madina. This is the most correct ruling as Allama Ramli says: Because the Holy Prophet's (Sallallahu alaihi wa sallam) migrated to Madina and His sanctuary also is in Madina, the people of Madina are distinguished from others.
(Al Maus‘ooatul Fiqhiya Al Kuwaitiya, Vol 27, Pg No: 143)
It has been established by the rigorously-authenticated (Sahih) Hadith that Hadhrat ‘Umar (May Allah be well pleased with him), Hadhrat ‘Uthman (May Allah be well pleased with him), Hadhrat ‘Ali (May Allah be well pleased with him) used to offer 20 Raka’at of Taraweeh and in the eras of their caliphate, it was also the custom of the Muslim Ummah. More than that, the Sahabah had arrived at a consensus about this as understood by the explanations given by the Imams, Fuqaha, etc. In general, in the Muslim world including in Makka and Madina, this is what is followed.