This article is for Hanafi-Mazhab (more will be added)
Rennet is an enzyme used in the food industry for the preparation of cheese by curdling milk. Rennet extracted from the stomach of a calf or animal whose meat is Halaal (lawful) after slaughtering in accordance with Islamic Thabh is unanimously Halaal (permissible) for consumption. If the above mentioned animals were not slaughtered in accordance with Islamic law then the Honourable Saahibayn of the Hanafi Legal School, Imam Abu Yusuf and Imam Muhammad, hold the view that the rennet itself is Halaal but its consumption would be Haraam (unlawful) because of its contamination with impure moisture etc. found in the stomach of the animal. According to them, rennet is fluid or viscous [running] and because of the penetration and saturation of impure moisture it would be impossible to purify it, therefore such rennet would remain Haraam.

However, Imaam Abu Hanifah does not regard the moisture found in the stomach of the above mentioned animals to be impure. Therefore the rennet obtained from them is Halaal, according to him.

The permissibility of rennet, therefore, is not really dependent on the slaughterer being Muslim or non-Muslim but upon whether there is life in rennet or not, and whether it absorbs the impurities found in the stomach of a newly born slaughtered calf. This is the outcome of the difference between Imaam Abu Hanifah and his two students, the Saahibayn,

Because of this difference in the opinion of the school, the Fatwa (legal opinion) allows a person to consume rennet (in the form of cheese etc.). Whilst Taqwa (piety) desires abstinence.

Allah Knows Best.

Further clarification and authentification.

1) In a commentary of the hadith text, Mishqatul Masaabeeh, by the famous Hanafi, scholar, Ali Qaari, Mirqaat, volume 8, page 193, is mentioned in the Kitaabul A’timah, Fasl Thaani.

Ibn Umar, radhiallahu anhu, narrates that our Rasul, sallallahu alayhi wasallam, was given a piece of Jubnah [Cheese] in Tabuk; he called for a knife, praying the tasmiyya he cut it [and ate it] narrated by Abu Dawud.

In the commentary, is mentioned; in this is proof of the purity of anfahah, because if it was impure then the cheese itself would be impure for it cannot be made without it [anfahah].

Again In this commentary in volume 2, page 79/80 is mentioned, in Bab Mash alal Khufayn, Fasl Awwal.

Tabraani narrates with a good sanad, although Ghareeb, that in one of the ghazawah, cheese was brought to Nabi sallallahu alayhi wasallam; He asked where it was made? Someone replied in Persia or in the lands of the Majus. Place the knife on it [cut it] and eat it. Someone said; O Rasulullah, we fear it may be carrion [Maitah]. He replied pray the tasmiyyah and eat it. Tirmidhi has mentioned a hadith wherein the Prophet was given a pair of leather socks and he wore them, without knowing whether they were pure or not. In the hadith of Salmaan, our Rasul was questioned regarding Jubn [cheese], Simn [clarifed butter] and Faraa’e [leather] together with the fact that it was taken from the lands of the Majus. It was mentioned to Umar regarding Jubn [cheese] and said to him that the rennet of dead animals are put in it [cheese]. He replied pray the tasmiyyah and eat it. Imam Ahmad mentions that this is the most sound hadith regarding cheese manufactured by Majus.

From the above commentaries it is evident that if rennet is derived from an animal that is permissible to eat, despite how it is slaughtered, it is Halaal to eat such cheese, despite the method used in the making of cheese.The rennet formed by the milk drunk by a newly born calf, which is then slaughtered and taken from its stomach is somewhat solid and has the effect of solidifying liquid milk and transforming it into cheese. Its permissibility though seemingly irrational because what is within the womb and stomach is under the law of impurity but the permissibility of cheese is established by Nass [Sharii proof] and agreed, therefore rennet is pure and permissible Fatwa Mazahir Ulum Volume 1 page 110.

2) Mufti Taqi’ Uthmaani, mentions that the ruling of Hadhraat Saahibayn is more cautious whilst that of Imaam Abu Hanifah is extensive or accommodating. If a pious person refrains it is better, whilst if a person is consuming cheese do not stop him. However, the rennet derived from the stomach of a pig is Haraam and Impure, provided its original property and chemical makeup does not undergo any drastic change.

Vegetarian or synthetic rennet, if nothing impure has been added to it will be Halaal. The cheese sold in supermarkets and shops, if it is vegetarian or derived from animals slaughtered Islamically is permissible to consume. On the other hand if it is pig rennet used in the cheese, without transformation then this is not permissible. Lastly, the rennet in cheese from animals other than pig is permissible, but abstention is desirable.