Note that among the "founders" of the four sunni madhabs (you quoted, as one should add the dhahiri school, which still exists to them too) only Imam a-Shafi'i really quoted and recorded his osol (sources or principles) of fiqh himself in his book ar-Risala (and also in his records like kitab jamma'a al-'ilm, ikhtilaf al-Hadith, ahkam al-Quran and ibtal al-Istihssan) which is the first book about osol al-fiqh.
The students of Imam abu Hanifa, Malik and Ahmad and their students etc. concluded the osol of the madhab based on statements of the scholars and based on their transmitted relief (for example musnad abu Hanifa and kitab al-Kharaj ... in case of the hanafi madhab and al-Muwatta' and al-Mudawana ... in case of the maliki madhab ... musnad Ahmad and Masaa'il Ahmad ... in case of the hanbali madhab.).
So even if there are reports saying (or pretending) that some muslim scholars have recorded books or papers on the matter before or meanwhile the earliest (written) books on the matter we have in hand now are (as far as i can tell):
- from the hanafi school of fiqh from al-Karkhi (died 340 a.H.),
- from the maliki school ibn al-Qassar (died 398 a.H.) and
- for the hanbali school ibn Ya'ala (died 458 a.H.).
Note that maliki scholars are known for their wide and big efforts in defining and extracting the goals of shari'a مقاصد الشريعة from the sources, while hanbali scholars were well known for their efforts in collecting narrations and fighting bidah. Ibn Kathir said:
يصعب على المبتدع أن يعيش في مكان يكثر فيه الحنابلة
It is hard for people who do inovations to live in a place where there are hanabilah.
this might explain to some extent the late compilation of the osol.
Also note the science of osol al-fiqh has changed a lot afterwards one could say it has been improved by works of scholars like imam al-Ghazali (shafi'i) and abu Zayd a-Dabusi (hanafi). Now let's see the osol for each madhab with details where i could find some:
The sunnah: Here they accept khabar al-ahad (simply said: a hadith which has at most three different narrators, what is more than that is called mashhur and if there are many narrators we call it sahih) under conditions:
first the rawy (narrator) himself shouldn't do anything opposed to what was narrated on his authority. For example lets say Anas said one could start eating before traveling in Ramadan but he only just started once he was on the road, than such a narration would be rejected.
Secondly if it isn't something which may cause any kind of harm.
Thirdly if it doesn't oppose qiyas.
- The saying of sahaba or fatwa of sahaba: abu Hanifa accepted only opinions of sahaba, as he used to say if it comes to an opinion of tabi'yn than the opinion of anybody is not better or worse than mine and it is not binding.
Trickery: For example if you knew Ramadan entered and didn't want to fast and traveled to be excused from fasting (here's an other example of this). Note this is an opposite source to what hanabilan and malikiya call sadd a-dhraai'i سد الذرائع (closing any path that may guide to something which may cause harm). There are three given categories of tricks:
Trick of a munafiq who pretends to be a Muslim this is an illegal or totally canceled trickery.
If a Muslim says something which declares him as a kafir (because of fear) while his heart is full of faith. This is considered as not canceled and therefore allowed trickery.
And all of what might neither be the first kind nor the second and those are the reason of disagreement between the hanafis and other schools. Therefore scholars of the hanafi school have divided them into: allowed, haram, recommended,
frowned upon and essential. And a trick in shari'a is defined as anything which is free from haram or something which is not allowed and doesn't let you commit a sin.
- Customs and shari'a of former nations (this is quoted in the Arabic wikipedia instead of trickery).
Sources wikipedia AR, EN and this Arabic post.
They usually divide their osol into two or three kinds: transmitted sources (based on texts) and intellectual sources (where one needs to use his mind to apply a ruling which is not clearly in a text) and some add the view of objectives -> goals of shari'a.
The transmitted sources are: Quran, Sunnah, Ijma', Ijam' ahl-al-Madina, Saying and fatwas of sahaba, Costums (to some extent)
- The Quran
- The sunnah: As far as i can tell the maliki consider khabar al-ahad (especially if it is a only hadith with an order or prohibition etc., not a hadith describing an act) as a weak or less reliable hadith, as long as it is no support in the customs of people of Medina.
Ijma' (here Imam Malik made clear that what is defined as consensus is the consensus of scholars
"وما كان فيه الأمر المجتمع عليه فهو ما اجتمع عليه أهل الفقه والعلم ولم يختلفوا فيه"
And that what is called consensus is the matter on which people of Knowledge and wisdom have been unified and didn't vary from.
Ijma' ahl al-Madina (or more correct the customs of people of Medina) (transmitted source): Note that Imam Malik sometimes differed from it if this customs or consensus was based on ahad, this means only if this custom has been accepted or approved by many or the majority of scholars it had a higher status than an ahad hadith.
Note that this source was objected by many scholars like al-Laith ibn Sa'ad and a-Shafi'i, who showed many cases where Malik even objected the consensus of people of Medina. But if we look in the details of the dispute we may find that one could say it is much ado about nothing. It was also discussed among the scholars of the maliki school itself, as there are different views about it.
- Saying or fatwa of a sahabi
- al-Maslaha al-mursala is often translated public interest (See the link at the end of the post for details) the only condition for this to be applied is that it shouldn't contradict a clear text (Quran/sunnah). Note that many scholars like Imam a-Shafi'i have objected this, but in his doctorate sheikh al-Buty may Allah have mercy on him has shown that in almost all madhabs this is considered but with a different kind of definition in terms.
- Costums are directly related to Maslaha (benefit)
- Sadd a-dharaai'i
- Al-Istishab can be translated as an accompanying matter, this means that a status is kept until we have an evidence for a change. For example: One can assume that something stays a property of the owner until we have an evidence that the ownership has changed.
Malik also said that the shari'a of the former nations is a shari'a for us (as far that there's no clear text abrogating it).
And the malikis are some kind of open towards other schools as it allows a maliki to adopt acts or fatwas of other schools to prevent disapproval or useless discussion. For example a maliki can pray behind any other Muslim as long as he isn't known as fasiq, even if this Imam didn't apply one of the (maliki) conditions of prayer, for example he could pray behind a person who has slept and didn't do wudu' or a person who doesn't consider the takbir to start prayer (takbirat al-Ihram) a condition of validity of a prayer. And if there's no fatwa on a matter from maliki scholar a mufti could take a fatwa from an other madhab ...
The objective view is mostly related to the consideration of the benefit or public interest which is often quoted in the madhab.
Sources wikipedia AR EN
Note that Imam a-Shafi'i has changed his madhab when he went to Egypt, so we call anything which came before he settled down there the ancient madhab and anything afterwards the new madhab.
- The Quran
The sunnah in contradiction to the schools quoted above the later schools of fiqh consider all kinds of hadith as reliable, as long as they are sahih, in cases (hanbali) they also consider weak hadith as reliable. One could say that al-Muwatta' of Imam Malik was more likely binding for shafi'is than for malikis. A-Shafi'i wrote in ar-Risala the sunnah has the same status as Quran
على أن السنة منزَّلة كالقرآن،
based on (33:34)
And remember what is recited in your houses of the verses of Allah and wisdom. ...
He also divided sunnah into three categories:
First matters which have been revealed in the Quran and our Prophet explained them as it was in the Quran.
Secondly matters which have been addressed in the Quran, and our Prophet () made clear what Allah meant.
Thirdly matters which have been made law by our Prophet and there's no text in the Quran about them. Here scholars of the school had two opinions some said it was by divine decree as we are asked by Quran to follow our Messenger. Some said there's nothing that has no source in the Quran.
Al-Ijma' is defined as the consensus of the mujtahids of the ummah in a period of time. This is differs to the definition given by Imam Malik!
- Saying or fatwa of a sahabi (in the ancient madhab) he later rejected it and only allowed a consensus of sahaba in the meaning that none of them is known to have had objected a given fatwa on a matter given by an other or a majority of sahaba.
Note 1: When Imam a-Shafi'i wrote refuting istihsaan he meant something different than that what the hanafis applied and the malikis adopted from them. A-Shafi'i was speaking about a qualification between different opinions which is not based on a source, which is not the case for the definition adapted by the hanafis and malikis.
And also as said before when he refuted al-maslaha al murssala and used what he called al-munasabsa المناسبة instead which is concluded via qiyas.
Note 2: Honestly speaking if we take a deep look into several sources then most of the sources expect: Quran, Sunnah, Ijma', Ijma' a-sahaba, fatwa of sahabi, Ijma' ahl-al-Medina, customs (which includes all transmitted sources) can be considered as utilities to improve qiyas or other sources as a-Shafi'i did as stated above.
Sources wikipedia AR EN
- The Quran
- The sunnah note that Imam Ahmad used to allow taking any sahih hadith as an evidence or source. He also allowed murssal (a hadith where the tabi'i didn't mention the sahabi on whoms authorithy he is narrating which is generally considered da'if) and da'if hadith (according Imam Ahmad's own definition) but they are placed in the ranking behind the fatwa of a sahabi, but before Qiyas!
- Fatwa of a sahabi: maybe it's worth mentioning that this is different than the consensus of sahaba, as there are cases where sahaba had different opinions, in that case one is asked to chose the fatwa which seems closest to Quran and sunnah.
- Qiyas: in the hanbali madhab is only used in case of necessity.
If istishab etc. where not in need of Qiyas one could say that Qiyas is the last refuge or source of this madhab.
Sources wikipedia AR EN
The dhahiri school rejects qiyas and basically has only Quran, Sunnah and Ijma'a of sahaba and saying or fatwa of sahaba as a source. Imam ibn Hazm even compiled a book refuting qiyas showing that one could base all fatwas on given texts. They take the texts as is they don't try to read behind the lines that's why they call themselves dhahirya as ظاهر in Arabic means obvious.
See also wikipedia AR and EN
For more information also take a look at the following posts:
What is the difference between qiyas and istislah?
What are Ijma (izma) and qiyaas (kias) and when are they applied?
Difference between Ijma' on transmission and Ijma' on opinion
What is Ijmaa? How is it established?
See also this post in Arabic.