Bismillah.
Volume 3, Book 31, Number 209:
Narrated Um Al-Fadl bint Al-Harith (radiallaahu `anhu):
"While the people were with me on the day of 'Arafat they differed as to whether the Prophet (sallallaahu `alayhi wasallam) was fasting or not; some said that he was fasting while others said that he was not fasting. So, I sent to him a bowl full of milk while he was riding over his camel and he drank it."
Volume 3, Book 31, Number 210:
Narrated Maimuna (radiallaahu `anhaa):
The people doubted whether the Prophet (sallallaahu `alayhi wasallam) was fasting on the day of 'Arafat or not, so I sent milk while he was standing at 'Arafat, he drank it and the people were looking at him.
Volume 3, Book 31, Number 211:
Narrated Abu 'Ubaid:
(the slave of Ibn Azhar) I witnessed the 'Id with 'Umar bin Al-Kattab (radiallaahu `anhu) who said, Allah's Apostle (sallallaahu `alayhi wasallam) has forbidden people to fast on the day on which you break fasting (the fasts of Ramadan) and the day on which you eat the meat of your sacrifices (the first day of 'Id ul Fitr and 'Id ul-Adha).
Volume 3, Book 31, Number 212:
Narrated Abu Sa'id (radiallaahu `anhu):
The Prophet (sallallaahu `alayhi wasallam) forbade the fasting of 'Id-ul-Fitr and 'Id-ul-Adha (two feast days) and also the wearing of As-Samma' (a single garment covering the whole body), and sitting with one's leg drawn up while being wrapped in one garment. He also forbade the prayers after the Fajr (morning) and the 'Asr (afternoon) prayers.
Volume 3, Book 31, Number 213:
Narrated Abu Huraira (radiallaahu `anhu):
Two fasts and two kinds of sale are forbidden: fasting on the day of 'Id ul Fitr and 'Id-ul-Adha and the kinds of sale called Mulamasa and Munabadha. (These two kinds of sale used to be practiced in the days of Pre-lslamic period of ignorance; Mulamasa means when you touch something displayed for sale you have to buy it; Munabadha means when the seller throws something to you, you have to buy it.)
Volume 3, Book 31, Number 214:
Narrated Ziyad bin Jubair:
A man went to Ibn 'Umar (radiallaahu `anhu) I. and said, "A man vowed to fast one day (the sub-narrator thinks that he said that the day was Monday), and that day happened to be 'Id day." Ibn 'Umar (radiallaahu `anhu) said, "Allah orders vows to be fulfilled and the Prophet (sallallaahu `alayhi wasallam) forbade the fasting on this day (i.e. Id)."
Volume 3, Book 31, Number 215:
Narrated Abu Said Al-Khudri (radiallaahu `anhu):
(who fought in twelve Ghazawat in the company of the Prophet (sallallaahu `alayhi wasallam)). I heard four things from the Prophet (sallallaahu `alayhi wasallam) and they won my admiration. He said;
1. "No lady should travel on a journey of two days except with her husband or a Dhi-Mahram;
2. "No fasting is permissible on the two days of Id-ul-Fitr and 'Id-ul-Adha;
3. "No prayer (may be offered) after the morning compulsory prayer until the sun rises; and no prayer after the 'Asr prayer till the sun sets;
4. "One should travel only for visiting three Masajid (Mosques): Masjid-ul-Haram (Mecca), Masjid-ul-Aqsa (Jerusalem), and this (my) Mosque (at Medina)."
Volume 3, Book 31, Number 216:
Narrated 'Aisha and Ibn 'Umar (radiallaahu `anhu):
Nobody was allowed to fast on the days of Tashriq except those who could not afford the Hadi (Sacrifice).
Volume 3, Book 31, Number 217:
Narrated Ibn 'Umar (radiallaahu `anhu):
Fasting for those who perform Hajj-at-Tamattu' (in lieu of the Hadi which they cannot afford) may be performed up to the day of 'Arafat. And if one does not get a Hadi and has not fasted (before the 'Id) then one should fast of the days of Mina. (11, 12 and 13th of Dhul Hajja).
Volume 3, Book 31, Number 218:
Narrated Salim's father:
The Prophet (sallallaahu `alayhi wasallam) said, "Whoever wishes may fast on the day of 'Ashura'."
Volume 3, Book 31, Number 219:
Narrated 'Aisha (radiallaahu `anhaa):
Allah's Apostle (sallallaahu `alayhi wasallam) ordered (the Muslims) to fast on the day of 'Ashura', and when fasting in the month of Ramadan was prescribed, it became optional for one to fast on that day ('Ashura') or not.
Volume 3, Book 31, Number 220:
Narrated Aisha (radiallaahu `anhaa):
Quraish used to fast on the day of 'Ashura' in the Pre-lslamic period, and Allah's Apostle (sallallaahu `alayhi wasallam) too, used to fast on that day. When he came to Medina, he fasted on that day and ordered others to fast, too. Later when the fasting of the month of Ramadan was prescribed, he gave up fasting on the day of 'Ashura' and it became optional for one to fast on it or not.
Volume 3, Book 31, Number 221:
Narrated Humaid bin 'Abdur Rahman:
That he heard Muawiya bin Abi Sufyan (radiallaahu `anhu) on the day of 'Ashura' during the year he performed the Hajj, saying on the pulpit, "O the people of Medina! Where are your Religious Scholars? I heard Allah's Apostle (sallallaahu `alayhi wasallam) saying, 'This is the day of 'Ashura'. Allah has not enjoined its fasting on you but I am fasting it. You have the choice either to fast or not to fast (on this day).' "
Volume 3, Book 31, Number 222:
Narrated Ibn 'Abbas (radiallaahu `anhu):
The Prophet (sallallaahu `alayhi wasallam) came to Medina and saw the Jews fasting on the day of Ashura. He asked them about that. They replied, "This is a good day, the day on which Allah rescued Bani Israel from their enemy. So, Moses fasted this day." The Prophet (sallallaahu `alayhi wasallam) said, "We have more claim over Moses than you." So, the Prophet (sallallaahu `alayhi wasallam) fasted on that day and ordered (the Muslims) to fast (on that day).
Volume 3, Book 31, Number 223:
Narrated Abu Musa (radiallaahu `anhu):
The day of 'Ashura' was considered as 'Id day by the Jews. So the Prophet (sallallaahu `alayhi wasallam) ordered, "I recommend you (Muslims) to fast on this day."
Volume 3, Book 31, Number 224:
Narrated Ibn 'Abbas (radiallaahu `anhu):
I never saw the Prophet (sallallaahu `alayhi wasallam) seeking to fast on a day more (preferable to him) than this day, the day of 'Ashura', or this month, i.e. the month of Ramadan.
Volume 3, Book 31, Number 225:
Narrated Salama bin Al-Akwa (radiallaahu `anhu):
The Prophet (sallallaahu `alayhi wasallam) ordered a man from the tribe of Bani Aslam to announce amongst the people that whoever had eaten should fast the rest of the day, and whoever had not eaten should continue his fast, as that day was the day of 'Ashura'.
The categories of rulings are five: obligatory (waajib), forbidden (muharram), encouraged (mustahabb), disliked (makrooh) and permissible (mubaah).
1. Obligatory fasts
(i) The Ramadaan fast
(ii) Making up missed Ramadaan fasts
(iii) Expiatory fasts (expiation for accidental killing, expiation for zihaar (a form of jaahili divorce), expiation for having intercourse during the day in Ramadaan, and expiation for breaking an oath)
(iv) Fasting for the pilgrim who does tamattu’ in Hajj if he does not have a sacrificial animal. “and whosoever performs the ‘Umrah in the months of Hajj, before (performing) the Hajj, (i.e. Hajj-at-Tamattu‘ and Al-Qiraan), he must slaughter a Hady such as he can afford, but if he cannot afford it, he should observe Sawm (fasts) three days during the Hajj and seven days after his return (to his home)” [al-Baqarah 2:196 – interpretation of the meaning].
(v) Fasting in fulfilment of a vow
2. Mustahabb (encouraged) fasts
(i) Fasting the day of ‘Ashoora’
(ii) Fasting the day of ‘Arafah
(iii) Fasting on Mondays and Thursdays each week
(iv) Fasting three days of each month
(v) Fasting six days of Shawwaal
(vi) Fasting most of the month of Sha’baan
(vii) Fasting the month of Muharram
(viii) Fasting alternate days – which is the best of fasting
3. Makrooh (disliked) fasts
(i) Singling out Friday for fasting – because the Prophet (peace and blessings of Allaah be upon him) said: “Do not fast on a Friday unless you fast a day before or a day afterwards.” Agreed upon
(ii) Singling out Saturday for fasting – because the Messenger(peace and blessings of Allaah be upon him) said: “Do not fast on Saturdays apart from days when you are obliged to fast, even if one of you cannot find anything other than grape stalks or the bark of a tree (to suck on, to make sure that he is not fasting).” Narrated by al-Tirmidhi, 744, who classed it as hasan. Also narrated by Abu Dawood, 2421; Ibn Maajah, 1726; classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 960.
Al-Tirmidhi said: What is meant by its being makrooh is that a man should not single out Saturday for fasting because the Jews venerate Saturday. End quote.
4. Forbidden fasts
(i) Fasting on Eid al-Fitr, Eid al-Adha and the days of Tashreeq, which are the three days after Eid al-Adha.
(ii) Fasting on the “day of doubt” – which is the thirtieth of Sha’baan, if the sky was cloudy and the new moon could not be sighted. But if the sky was clear there can be no doubt.
(iii) Fasts observed by women who are menstruating and bleeding following childbirth.
5. Permissible fasts – these are fasts that do not come under any of the four headings mentioned above.
What is meant by permissible here is that there is no report enjoining or forbidding fasting on this day in particular, such as Tuesdays and Wednesdays, even though in principle, observing a voluntary fast is an act of worship that is encouraged.
See al-Mawsoo’ah al-Fiqhiyyah, 28/10-19; al-Sharh al-Mumti’, 6/457-483