The Qur'an is the first root and is not in dispute. The sunna is universally accepted in principle, but the schools differ sharply over which reports qualify, and over whether a report transmitted by a single chain may restrict a general Qur'anic statement, a question on which the Hanafi and Shafi'i traditions divide.
Ijma, consensus, is accepted as binding by all four Sunni schools, though what body must agree was never fully settled: the Companions, the scholars of a given age, or the whole community have all been proposed, and later jurists conceded that consensus after the first generations is almost impossible to verify. Twelver Shia jurisprudence accepts consensus in a restricted sense, valuing it because it is taken to disclose the view of the Imam rather than because agreement is authoritative in itself.
Qiyas, analogy, is the fourth root for the four Sunni schools: a ruling is extended from a text-governed case to a new one sharing its operative cause. It is rejected outright by the Zahiris, whose most formidable exponent, Ibn Hazm, held that extending the law by analogy is legislating; and Twelver Shia jurisprudence rejects qiyas in its Sunni form, putting aql, reason, in its place.
Beyond these lie the disputed roots, and they are where the schools show their character. Istihsan, juristic preference, permits departure from a strict analogy where it produces hardship or absurdity; it is a Hanafi and Maliki instrument, and al-Shafi'i attacked it as deciding by taste. Maslaha, public interest, the Maliki istislah, and al-Shatibi's maqasid al-shari'a with its five aims of preserving religion, life, intellect, lineage and property, has become the central instrument of modern reformist argument, and the central worry of its critics, who ask what constrains an appeal to benefit. Istishab, the presumption that a known state continues until change is proved, is emphasised by Shafi'is and Hanbalis and underwrites the presumption of permissibility.
Within Twelver Shia jurisprudence the deepest dispute was whether these instruments were legitimate at all. The Akhbari movement of the seventeenth and eighteenth centuries argued for reliance on the reports of the Imams and against ijtihad and reason; the Usuli position, which defended both, prevailed by the late eighteenth century, and the modern structure of Shia legal authority rests on that outcome.