Firstly: 

Al-Mufassal refers to the short soorahs of the Qur’aan which have many breaks between them, in which the Basmalah (“Bismillah al-Rahmaan al-Raheem”) appears. They are called mufassal (divided) because a lot of breaks appear in them. 

There was a difference of scholarly opinion concerning the definition of al-Mufassal. Does it begin from Soorat Qaaf or from al-Hujuraat? There was also a difference of opinion as to which soorahs are regarded as long, medium and short. There are several opinions: 

In al-Mawsoo‘ah al-Fiqhiyyah (33/48) it says: They differed concerning al-Mufassal. The Hanafis are of the view that the long soorahs of al-Mufassal start with al-Hujuraat and end with al-Burooj; the medium soorahs begin there and end with al-Bayyinah; and the short soorahs begin there and go to the end of the Qur'aan. 

The Maalikis said that the long soorahs of al-Mufassal start with al-Hujuraat and end with al-Naazi‘aat; the medium soorahs begin with ‘Abasa and end with al-Duha; and the short soorahs begin with al-Duha, until the end of the Qur'aan. 

The Shaafa‘is said: The long soorahs of al-Mufassal are soorahs such as al-Hujuraat, al-Qamar and al-Rahmaan; the medium soorahs are soorahs such as al-Shams and al-Layl; and the short soorahs are soorahs such as al-‘Asr and Qul Huwa Allaah Ahad (al-Ikhlaas), 

The Hanbalis are of the view that al-Mufassal begins with Soorah Qaaf, because of the hadeeth of Aws ibn Hudhayfah who said: I asked the companions of the Messenger of Allah (blessings and peace of Allah upon him) how they divided the Qur’aan. They said, Three and five and seven and nine and eleven and thirteen, and Hizb al-Mufassal [This refers to the number of soorahs in each hizb or portion of Qur’aan. Thus the first hizb contains the first three soorahs, the second contains the next five soorahs, and so on. Hizb al-Mufassal is the last portion of the Qur’aan, from Soorat Qaf until the end]. Narrated by Abu Dawood (1396) and Ibn Maajah (1345); its isnaad was classed as hasan by Ibn Katheer in Fadaa’il al-Qur’aan (83) and as da ‘eef (weak) by al-Albaani in Da‘eef Abi Dawood. They said: This implies that the beginning of al-Mufassal is the forty-ninth soorah from the beginning of al-Baqarah, not from al-Faatihah. The end of the long soorahs is al-Naba’, the medium soorahs go up to al-Duha and the short soorahs go from there to the end of the Qur’aan. End quote. 

See: Fath al-Baari, 2/249; al-Itqaan fi ‘Uloom al-Qur’aan by al-Suyooti, 1/180 

Al-Haafiz Ibn Hajar (may Allah have mercy on him) said: 

Al-Mufassal starts with Soorah Qaaf and goes to the end of the Qur’aan, according to the sound opinion. It is called mufassal because there many breaks between the soorahs, where the Basmalah appears, according to the correct opinion. End quote from Fath al-Baari, 2/295. See also Fath al-Baari, 9/43. 

Secondly: 

With regard to who first called that al-Mufassal, this name was widespread and well known among the Sahaabah and was narrated in many various hadeeths from the Sahaabah. 

A man came to Ibn Mas‘ood and said: I recited al-Mufassal last night in one rak‘ah. He said: This is like the hasty recitation of poetry. I know the pairs of soorahs that the Messenger of Allaah (blessing and peace of Allah be upon him) used to recite in each rak‘ah. And he mentioned twenty soorahs from al-Mufassal, two soorahs in each rak‘ah. 

Narrated by al-Bukhaari (775), Muslim (922). 

The hadeeths in which the name al-Mufassal is mentioned are very many and various, in al-Saheehayn and elsewhere, which makes it likely that this name was narrated from the Prophet (blessings and peace of Allah be upon him). 

It was narrated that Jaabir ibn ‘Abd-Allah (may Allah be pleased with him) said: Mu‘aadh used to pray with the Prophet (blessings and peace of Allah be upon him), then he would go back and lead his people in prayer. [One night] he prayed ‘Isha’ and started to recite Soorat al-Baqarah, and one man left the congregation (and prayed by himself). It was as if Mu‘aadh criticized him. News of that reached the Prophet (blessings and peace of Allah be upon him) and he said: “[You are] one who puts people off praying in congregation, one who puts people off praying in congregation, one who puts people off praying in congregation roblems,” three times, and he told him to recite two of the medium-length soorahs of al-Mufassal. ‘Amr - the narrator of the hadeeth - said: I do not remember them. 

Narrated by al-Bukhaari (701) and Muslim (465). 

And Allah knows best.