1- Six days in the month of Shawwal (after the month of Ramadan)
The Prophet ﷺsaid “Whosoever fasts the month of Ramadan and thereafter follows it up with six days of fasting in Shawwal, it will be recorded (altogether) for him as a complete year of fasting.” [ Source: Muslim.]
It is the same if he does it consecutively or spreads it over the entire month.
2- First nine days of Dhul Hijjah (the twelfth month of the Islamic calendar)
The Prophet ﷺ said : “There is no other day in which good deeds performed are so pleasing to Allah than these days (i.e. the first ten days of the twelfth month). The companions inquired: “What about jihad (fighting) in the cause of Allah?” The Prophet said: “Not even jihad in the cause of Allah (is up to its rewards), except for a man who went out in the cause of Allah with his life and wealth and did not return with anything of either (of them).” [ Source: Bukhari.]
However, the most important of these days is the day of ‘Arafah, for those not performing the pilgrimage. The day of ‘Arafah is the ninth day of the twelfth month of the Islamic calendar. The Prophet ﷺ said: “As for fasting on the day of ‘Arafah, I expect Allah Almighty will reward that by forgiving the sins of a year gone and a coming year.” [ Source: Muslim.]
3- Fasting on ‘Ashura
This is because the Prophet said: “And as for fasting on the day of ‘Ashura, I expect that Allah Almighty shall forgive the sins of the year prior to it.” [ Source: Muslim.]
The reason for fasting on this day is in accordance with what was related from the authority of ‘Abdullah ibn ‘Abbas, who said: “When the Prophet ﷺ arrived at Madinah, he found the Jews fasting on ‘Ashura. Then he inquired why (do they fast on that day)? They answered, ‘This a good day, the day Allah Almighty saved the children of Israel from their enemy, and as such Prophet Musa (peace be upon him) fasted thereafter (on that day).’ Then the Prophet said: ‘I have more right to Musa than you.’ Then he fasted and ordered the Muslims to fast on that day.” [ Source: Bukhari.] In another narration by Imam Muslim, the Prophet said, “If I remain (alive) till the next year, I shall fast the ninth day.” [ Source: Muslim.]
4- The bright days in every month
It is authenticated that Abd al-Malik ibn al-Minhal conveyed from his father that the Prophet ﷺ ordered them to fast the three “bright” days. He said, “It is [equivalent] to fasting the entire month.” [ Source: Ibn Habban.]
5- Fasting on Monday and Thursday of every week
Abu Hurayrah reported that the Prophet ﷺ said: “Deeds are reported to Allah Almighty on every Monday and Thursday; and I like that my deeds are reported while I am fasting.” [ Source: Tirmidhi.]
6- Fasting on alternative days.
The best form of supererogatory fast is the fast of Prophet Dawud (peace be upon him). He used to fast a day and take a break the next day. Abdullah ibn ‘Amr reported the Prophet ﷺ to have said: “The best form of fasting is the fasting of Dawud (peace be upon him); he used to fast a day and take a break on the next (day).” [ Source: An-Nasaii.]
7- Fasting in Muharram (the first month of the Islamic calendar).
On the authority of Abu Hurayrah, the Prophet ﷺ said: “The best form of fasting besides Ramadan is in the month of Allah, Al-Muharram.” [ Source: Muslim.]
8- Fasting in Sha’ban (the eighth month of the Islamic calendar)
“O Prophet of Allah! I have not seen you fast in any month as you do in Sha’ban.” The Prophet replied: “That is the month, between Rajab and Ramadan, about which many people are not conscious, and deeds are returned to the Lord of the worlds (during it). Thus I love that my deeds are returned to Allah Almighty while I am fasting.” [ Source: An-Nasaii.]
Regarding the prohibition contained in the Prophet’s ﷺ hadeeth, where he said: “When Sha’baan is half over, do not fast until Ramadan,” [ Source: Ibn Khuzaymah.] it either means that the prohibition is related to deliberately fasting only after the month is halfway over, or to joining Sha’baan with Ramadan. However, a person who continues fasting the second half of Sha’baan after fasting during the first part of the month and does not join that month with Ramadan in fasting, then there is nothing wrong with his fasting.