When in prayer you recite a section of the Qur'an that includes a sajdah, what is the ruling on performing the sajdah? Is it obligatory to do? Or is it recommended? I'm pretty sure there are madhahib-level differences, so in your answers please go into those with evidence.
I will restrict this to the hukm of Sajdah tilawah only. Since, there is a difference of opinion also on where/when it must be done.
Quoting Bidayat al-Mujtahid wa Nihayat al-Muqtasid:
About the hukm of the prostrations of recitation, Abu Hanifa and his disciples said that they are obligatory. Malik and al-Shafi'i said that they are prescribed as a sunna and are not obligatory. The reason for disagreement arises from their dispute about the implication of the commands requiring prostrations and the traditions that convey the meaning of the commands. For example, whether the (following) words of the Exalted:
"When the revelations of the Beneficient were recited unto them, they fell down, adoring and weeping" (Qur"an 19:58),
are to be interpreted as an obligation or a recommendation. Abu Hanifa interpreted them in their apparent meaning of obligation, while Malik and al-Shafi'i followed the (interpretation of) the Companions, as they were the ones best grounded in the meaning of the command of law. Thus, it has been established that 'Umar ibn al-Khattab recited surat al-Sajda on a Friday. He descended (from the pulpit) and made a prostration and the people prostrated with him. On the next Friday, he recited it again and the people prepared for the prostration, so he said,
"Wait! Wait! Allah has not prescribed it for us, unless we want to do it".
They said that this occurred in the presence of the Companions, and no disagreement was transmitted from any of them. They were the ones who best knew the essence of the law. This is used in support of that view by those who rely on the opinion of a Companion when there is no other conflicting evidence. The disciples of al-Shafi'i argued on the basis of the tradition of Zayd ibn al-Thabit, who said.
"I used to read out the Qur'an for the Messenger of Allah (God's peace and blessings be upon him). Once I recited surat al-Hajj and he did not prostrate, and neither did we".
These jurists also argued on the basis of the report "that the Prophet (God's peace and blessings be upon him) did not prostrate during (the recitation of ) al-Mufassal Surahs", and also on the reports that he did prostrate during their recitation, a reconciliation between these implies that the prostrations are not obligatory. Each one of them reported what he saw, that is, those who said that he prostrated and those who said that he did not. Abu Hanifa, on the other hand, relied in this on the argument that the principle is to construe the commands as implying an obligation, and also those reports that amount to commands.
Source: Bidayat al-Mujtahid wa Nihayat al-Muqtasid (The Distinguished Jurist's Primer - Vol1) by Ibn Rushd
The hadith which is used for the opinion that it is not obligatory is the following:
Imam al-Bukhari recorded from Umar that he said: "O people, sometimes we recite the verse of Sajdah, so whoever makes a prostration he act correctly and there is no sin upon the one who does not do so" .
Source: Sujud-at-Tilawah outside Salah
There is difference of opinion among the schools on whether performing prostrations of recitation (سجدة التلاوة) is obligatory. The view of the Hanafis is that it is obligatory, their evidence includes:
The verse of the Quran where those who do not prostrate on hearing the Quran are denounced; condemnation is not done except on a forbidden action or on abstaining from what is obligatory.
فمالهم لايؤمنون وإذا قرئ عليهم القرآن لا يسجدون
So what is [the matter] with them [that] they do not believe, And when the Qur'an is recited to them, they do not prostrate [to Allah]?
The Hadith which says that it is obligatory without limiting it to only those who intend to perfom it.
السجدة على من سمعها وعلى من تلاها
The prostration is (obligatory) for one who hears it (the recitation) and for one who recites it.
The hadith which implies that it is commanded and hence obligatory:
قال رسول الله صلى الله عليه وسلم: إذا قرأ ابن آدم السجدة فسجد، اعتزل الشيطان يبكي، يقول: يا ويلي، وفي رواية يا ويله - أمر ابن آدم بالسجود فسجد فله الجنة، وأمرت بالسجود فأبيت فلي النار
The Messenger of Allah ﷺ said: When the son of Adam recites a Verse of prostration and prostrates, the Shaitan withdraws, weeping and saying: "Woe unto him" (and according to the report of Abu Kuraib: "Woe unto me") the son of Adam was commanded to prostrate and he prostrated, so Paradise will be his; I was commanded to prostrate and I refused, so the Fire is mine."
The view of the majority is that it is not obligatory but sunnah, their evidence has been given in Abdullah's answer.