I like to know if Imam Muslim (Sahih Muslim) and Imam Bukhari (Sahih Bukhari) followed any of the Madhabs i.e. Shaffai, Maliki, Hanifi, or Hanbuli?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Sheikh al-Islam Ibn Taymiay was once asked about whether Imams al-Bukhari, Muslim, Abu Dawood, al-Tirmizi, al-Nasiae, Ibn Majah, Abu Dawood, al-Tayalisi, al-Darumi, al-Bazzar, al-Dara Qutni, al-Baihaqi, Ibn Khuzaimah, and Abu Ya'la al-Musili were Mujtahid (scholar who has reached a level of knowledge whereby he can choose the opinion which he thinks is more appropriate and more supported by evidence from al-Qur'an and Sunnah hence not following a given Mazhab) or were they Muqallid (who follows a given Mazhab)?
He answered: Praise be to Allah, the Lord of the Worlds. As for al-Bukhari and Abu Dawood , they both are Mujtahid and great Imams in Fiqh.
As for Imams Muslim, al-Tirmizi, al-Nasaie, Ibn Majah, Abu Y'ala, al-Bazzar and others, they used to follow the method of Ahl-al-Hadith (adherent to Hadith). They were neither Muqallid (followers of a certain Imam) nor reached the level of Mujtahid at all. In fact, most of time they used to get inclined to the sayings of the Imams of Hadith such as Imams al-Shafie, Ahmad, Ishaq, Abu Uabaid and the like. Some of these scholars even specialized in the Mazhab of an Imam such as Imam Abu Dawood who has the specialization in the knowledge of Imam Ahmad. The above-mentioned Imams were inclined to the Fiqh of Ahl-al-Hijaz (area that contains Makkah, al-Madinah and their surroundings) more than the Fiqh of Ahl al- Iraq like Imam Abu Hanifah and al-Thawri .
As for Abu Dawood al-Tayalisi, he was a senior Imam. He was contemporary to Yahya bin Saeed al Qattan, Yazeed bin Haroon al-Wasity, Abdullah bin Dawood, Wakee bin al-Jarrah, Abdullah bin Idrees, Mu'adah bin Mu'adh, Hafs bin Ghyath, Abdur Rahman bin Mahdi and the others who were in the period of Imam Ahmad 's teachers.
However, all the Imams used to highly respect the Sunnah of the Prophet (Sallallahu Alaihi wa Sallam) though some of them like Wakee and Yahya bin Saeed were inclined to the Fiqh of Ahl al-Iraq, i.e. Abu Hanifa and al-Thawri . Some of them like Abdul Rahman bin Mahdi and others were inclined to the Fiqh of Ahl al-Madinah i.e. Imam Malik and the like.
Imam al-Baihaqi was the follower of the Fiqh of al-Shahfie. Imam al-Daraqutni also was inclined to the Fiqh of al-Shafi'e but his Ijtihad was stronger than the Ijtihad of Imam al-Baihaqi though al-Baihaqi also was a Mujtahid. Imam al-Daruqutni was more versed in knowledge and Fiqh than Imam al-Baihaqi .
Allah knows best.