In a Nutshell: Background
There are about 25 five Madani surahs (Al-Baqarah, Aal Imran, al-Nisaa, al-Ma'idah, al-Anfaal, al-Tawbah, al-Noor, al-Ahzab, Muhammad, al-Fath, al-Hujurat, al-Hadeed, al-Mujadilah, al-Hashr, al-Mumtahanah, al-Saff, al-Jumu'ah, al-Munafiqoon, al-Taghabun, al-Talaq, al-Tahreem, al-Mutafifeen, al-Qadr, al-Baynah and al-Nasr) and 89 Makki surahs (the rest of the surahs).
Scholars define the Makki (Meccan) and Madani (Medinan) surahs with three main definitions: Makki is what had been revealed in Mecca and Madani is what was revealed in Medina; Mekki is what speaks with the people of Mecca and Madani is what speaks to the people of Medina; Makki is what had been revealed before Hijrah (even if it was outside Mecca) and Madani is what was revealed after Hijrah (even if revealed in Mecca or outside Medina). (Zarkashi, al-Burhan, Vol. 1, p. 187, Suyuti, al-Itqan, Vol. 1, p. 36, Zurqani, Manahil al-Irfan, Vol. 1, p. 193)
The latter definition is more accurate and soundly accepted by most of the classical scholars of Ulum al-Qur'an. (Ibid)
There is a very hot debate over the Makki and Madani surahs because they were not identified by the Prophet (saw), rather there are based on a large number of contradicting (weak and sound) narrations as well as the ijtihad of scholars in identifying them. They also disagree based on their definition of each term, as clarified above, as well as the elements and the characteristics they suggest to identify which surah belongs to which category.
The later editor scholars (al-muhaqiqun) could identify three main categories: agreed upon Madani surahs (20 surahs), agreed upon Makki surahs (82 surahs) and disputed surahs (12 surahs).
Note: in most, or many, of the surahs there are one or more verses belong to the other category, but the identification here is for the overall and the main nature of the surah. They were organised in a poem by the six-century mufasir Abu al-Hasan al-Hassar and used and authorised by later scholars. (Suyuti, al-Itqan, Vol. 1, pp. 44-45) Medinan Surahs
There are twenty surahs mostly agreed upon by scholars based on some narration of companions, tabi'een and later narrators. These twenty surahs are the following:
Al-Baqarah, Aal Imran, al-Nisaa, al-Ma'idah, al-Anfaal, al-Tawbah, al-Noor, al-Ahzab, Muhammad, al-Fath, al-Hujurat, al-Hadeed, al-Mujadilah, al-Hashr, al-Mumtahanah, al-Jumu'ah, al-Munafiqoon, al-Talaq, al-Tahreem and al-Nasr. (Ibid, Zurqani, Manahil al-Irfan, Vol. 1, p. 198) Disputed Surahs
Most scholars disagree on 12 surahs, I will here do my best, based mostly on the works done by the best scholar evaluated these surahs ibn Ashur, to identify which surah could more likely belong to which category. But bear in mind, any surah could have one or more verses from the other category, but the rule here is for the overall nature. Al-Fatihah:
It is more likely a Makki surah as it was a part of prayers that was initiated in the first year of ba'thah. This is the opinion of the majority of scholars, as ibn Ashur reported:
وَهَذِهِ السُّورَةُ مَكِّيَّةٌ بِاتِّفَاقِ الْجُمْهُورِ Al-Raad:
"The majority (of scholars) agree this is a Makki surah." (Ibn Ashur, al-Tahreer wa al-Tanweer, Vol. 1, p. 136)
It is also more likely a Makki surah with some Madani verses (such as verse 43), because the overall structure of the surah is highly similar to the Makki surahs. It is also a Makki surah as reported in sound narrations on the authority of ibn Abbas, Mujahid, Abu Talahah, Qatadah, Sa'id ibn Jubayr and others. The reason behind their disagreement here is generally because of weak and fabricated narrations. (Ibid, Vol. 13, p. 76) Al-Rahman:
It is also a Makki surah according to the opinion of the majority of companions and Tabi'een as well as the vast sound narrations documented its cause of revelation. Ibn Ashur noted:
وَالْأَصَحُّ أَنَّهَا مَكِّيَّةٌ كُلَّهَا ... وَهِيَ مِنْ أَوَّلِ السُّوَرِ نُزُولًا Al-Saff:
"The more likely view is that is a totally Makki surah … and it is one of the first revelations (surahs)." (Al-Tahreer wa al-Tanweer, Vol. 27, pp. 227-228)
It is an overall Madani surah contains a number of Makki verses as supported by the structure of the surah and some other narrations, as also noted by ibn Atiyah and ibn Ashur:
وهي مدنية في قول الجمهور… ويشبه أن يكون فيها المكي والمدني Al-Taghabun:
"It is a Madani surah according to the majority (of scholars) … it seems it contains the Makki and Madani (verses) as well." (Ibn Atiyah, al-Muharar al-Wajeez, Vol. 5. p. 301, Ibn Ashur, Vol. 28, p. 172)
It is a Madani surah according to the majority of mufasiroon, but its beginning (V. 1-13) is Makki according to ibn Atiyah and others. (Ibn Ashur, Vol. 28, p. 258, Ibn Atiyah, Vol. 5, p. 317) Al-Mutafifeen:
There are three main views regarding this surah, either it was the last surah revealed in Mecca, or the first surah revealed in Medina or on the route to Medina. Ibn Atiyah argued the former and ibn Ashur argued the later. But it seems it was revealed in Medina as it contains the story of a trader who cheats the scale which is the main story in the surah after which it was named. (Ibn Ashur, Vol. 30, p. 187, Ibn Atiyah, Vol. 5, p. 449) Al-Qadr:
One can't decide exactly where it was revealed, there are two narrations of ibn Abbas (ra) support both categories. Others, such as al-Waqidi, noted it was revealed in Medina after the legislation of fasting urging Muslims to pray the prayer in the day of al-Qadr. I personally find the latter argument and the context of the surah (that speaks to a believing community) could give weight to the Medinan notion. (Ibn Ashur, Vol. 30, p. 455, Ibn Atiyah, Vol. 5, p. 504) Al-Bayinah:
The majority of mufasiroon argue it is a Makki surah, but there is a sound narration mentions clearly the context after the revelation, as the Prophet (saw) was asked to recite it before ibn Ubbay who met the Prophet (saw) only in Medina:
لَمَّا نَزَلَتْ: لَمْ يَكُنِ الَّذِينَ كَفَرُوا مِنْ أَهْلِ الْكِتابِ إِلَى آخِرِهَا قَالَ جِبْرِيلُ: يَا رَسُولَ اللَّهِ إِنَّ اللَّهَ يَأْمُرُكَ أَنْ تُقْرِئَهَا أُبَيًّا
"When it was revealed, Gabriel said: O Messenger of Allah, Allah orders you to recite it to ibn Ubbay." (Ibn Ashur, Vol. 30, p. 467)
The context of the surah also shows it discusses some issues with the people of the book (Ahlu al-Kitab) which is one of the characteristics of Madani surahs. (Ibid) Al-Zalzalah:
The majority of the companions, such as ibn Abbas and ibn Mas'ood, and other tabi'een, such as Mujahid, Ataa and Dahaq, report it is a Makki surah. But Qatadah, Muqatil, ibn Atiyah argued it, or only the last verse, is a Madani surah based on a very weak narration. (Ibid) Al-Ikhlas:
There are narrations report the Prophet (saw) was asked by either a group of Quraysh in Mecca or a group of rabbis in Medina to describe his God, so the surah was revealed.
But the majority of scholars argue it was revealed in Mecca because of the sound narrations report it was revealed in Mecca and the Meccan characteristics, as well as the Prophet (saw), has to have presented his God in the first 13 years.
Some scholars such as Suyuti argued it was revealed twice, but if it was revealed in Mecca it would be only used in the discussion in Medina i.e. a usage of the surah not a new revelation that the narrator thought it was the first time to be revealed. (Ibn Ashur, Vol. 30, p. 611, Ibn Atiyah, Vol. 5, p. 536) Al-Mu'awizatayn (Al-Nas and Al-Falaq):
The surahs were used as purifying verses against any type of evil in the world. There are also two narrations. Some reports note it was revealed after the Prophet (saw) was harmed by a magician (Lubayd ibn al-A'sam) in Medina and some more reliable narrations of ibn Abbas (ra) report Quraysh wanted to harm the Prophet (saw) by evil eyes and the surahs were revealed to protect him.
The former narrations seem to apply more to the case as ibn Ashur suggested. (Ibid, Vol. 30, p. 624)
Thus, the Madani surahs from these disputed surahs are more likely: al-Saff, al-Taghabun, al-Mutafifeen, al-Qadr and al-Baynah; but the Makki surahs from them are al-Fatihah, al-Raad, al-Rahman, al-Zalzalah, al-Iklas, al-Falaq and al-Naas. Meccan Surahs
The Mekki surahs are the rest 82 surahs in addition to the above Makki surah evaluated from the disputed ones = 89 surahs. Conclusion
There are about 25 five Madani surahs (Al-Baqarah, Aal Imran, al-Nisaa, al-Ma'idah, al-Anfaal, al-Tawbah, al-Noor, al-Ahzab, Muhammad, al-Fath, al-Hujurat, al-Hadeed, al-Mujadilah, al-Hashr, al-Mumtahanah, al-Saff, al-Jumu'ah, al-Munafiqoon, al-Taghabun, al-Talaq, al-Tahreem, al-Mutafifeen, al-Qadr, al-Baynah and al-Nasr. ) and 89 Makki surahs (the rest of the surahs). References
Zarkashi, al-Burhan fi Ulum al-Qur'an
Suyuti, al-Itqan fi Ulum a,-Qur'an
Zurqani, Manahil al-Irfan fi Ulum a,-Qur'an
Ibn Ashur, al-Tahreer wa al-Tanweer
Ibn Atiyah, al-Muharar al-Wajeez