Fasting outside of Ramadan can be of two kinds: voluntary and obligatory fasting.
First: Voluntary fasting:
It is one of the acts of worship that expiate one's sinsaccording to the Hadeeth (narration) on the authority of Huthayfah, may Allah be pleased with him: "The afflictions caused for man by his wife, money and neighbor are expiated by his prayers, fasting and charity." It was also narrated on the authority of Abu Qataadah Al-Ansaari, may Allah be pleased with him, that the Prophet , was asked about fasting the Day of ‘Arafah (9th of ThuI-Hijjah), whereupon he said: “It expiates the sins of the preceding year and the coming year.” He was also asked about fasting the Day of ‘Aashooraa’ (10th of Muharram), whereupon he said: “It expiates the sins of the preceding year." [Muslim] In addition, Abu Sa‘eed Al Khudri, may Allah be pleased with him, narrated that the Prophet, , said: "Anyone who fasts one day for the sake of Allah, Allah will keep him away from Hell for a distance covered in seventy years." [Al Bukhari and Muslim] Abu Umaamah, may Allah be pleased with him, narrated that the Prophet, , said: "Whoever fasts a day for the sake of Allah, Allah will place between him and Hell a trench like that between the heavens and the earth."[At-Tirmithi]
Voluntary fasting can be of two types:
A- Free fast:
Al-Bukhari and Muslim narrated on the authority of Anas, may Allah be pleased with him, that the Messenger of Allah, , used to fast until one would say that he would never stop fasting, and he would abandon fasting until one would say that he never fasted. Moreover, they narrated on the authority of Ibn ‘Abbaas, may Allah be pleased with him, that the Prophet, , never fasted a full month except the month of Ramadan, and he used to fast till one could say, "By Allah, he will never stop fasting," and he would abandon fasting till one would say, "By Allah, he will never fast."
B- Specific fast:
1- Fasting of ‘Aashooraa’
In a Hadeeth on the authority of Abu Moosa Al Ash‘ari, may Allah be pleased with him, he said that the Day of ‘Aashooraa’ was one which the Jews respected and treated as a festival. The Messenger of Allah, , said: "I recommend you (Muslims) to observe fast on this day." [Al-Bukhari and Muslim]
Ibn ‘Abbaas, may Allah be pleased with him, narrated that when the Messenger of Allah, , fasted on the Day of ‘Aashooraa’ and commanded that it should be observed as a fast, they (his Companions) said to him, ‘O Messenger of Allah, it is a day which the Jews and Christians glorify.’ Thereupon, the Messenger of Allah, , said: "Next year, Allah willing, I will fast the ninth (of Muharram)." However, the Messenger of Allah, , died before the advent of the next year. [Muslim]
2- Fasting the Day of ‘Arafah
It was narrated on the authority of Abu Qataadah, may Allah be pleased with him, that the Prophet, , was asked about fasting the Day of ‘Arafah (9th of Thul-Hijjah), whereupon he said: “It expiates the sins of the preceding year and the coming year.” He was asked about fasting the Day of ‘Aashooraa’ (10th of Muharram), whereupon he said: “It expiates the sins of the preceding year." [Muslim]
3- Fasting six days of Shawwaal:
Abu Ayyoob Al-Ansaari, may Allah be pleased with him, narrated that the Messenger of Allah, , said: "Whoever fasts Ramadan and then follows it with six (fasts) of Shawwal, it would be as if he fasted all his life." [Muslim]
4- Fasting nine days of Thul-Hijjah
It was narrated on the authority of Hunaydah ibn Khaalid, may Allah be pleased with him, the stepson of ‘Umar ibn Al Khattaab, may Allah be pleased with him, that a woman heard Umm Salamah, may Allah be pleased with her, the wife of the Prophet, , saying, “The Messenger of Allah used to fast nine days of Thul-Hijjah and three days of each month – the first Monday and two Thursdays.”[Abu Daawood and An-Nasaa’i]
Ibn Abbaas, may Allah be pleased with him, narrated that the Prophet, , said: "‘No good deeds done on other days are superior to those done on these (first ten days of Thul-Hijjah).’ They (the Companions) said, ‘Not even Jihaad?’ He replied: ‘Not even Jihaad, except that of a man who does it by putting himself and his property in danger (for the sake of Allah) and does not return with either of those." [Al-Bukhari]
5- Fasting in Muharram
Abu Hurayrah, may Allah be pleased with him, narrated that the Messenger of Allah, , said: "The best fast after Ramadan is the fast in the month of Allah Al-Muharram and the best prayer after the obligatory prayer is the prayer offered at night." [Muslim]
6- Fasting in Sha‘baan
‘Aa’ishah, may Allah be pleased with her, said, "I never saw the Prophet fasting (voluntarily) more in any other month than that of Sha‘baan. He used to fast throughout the whole of Sha‘baan except for a few (days)." [Al-Bukhari and Muslim] In addition, Usaamah ibn Zayd, may Allah be pleased with him, said, "I said, 'O Messenger of Allah, I do not see you fasting in any other month like you fast in Sha‘baan.' He said: 'That is a month to which people do not pay attention, between Rajab and Ramadan, and it is a month in which deeds are lifted up to the Lord of the Worlds. I like my deeds to be lifted up when I am fasting.'” [An-Nasaa’i]
7- Fasting certain days every week
’Aa’ishah, may Allah be pleased with her, narrated that the Prophet, , used to fast Mondays and Thursdays [An-Nasaa’i, At Tirmithi and Ibn Maajah]
8- Fasting the White Days of every month
It was narrated on the authority of Milhaan, may Allah be pleased with him, that the Prophet, , used to order them to fast the White Days; the thirteenth, fourteenth and fifteenth of every lunar month and he said: "They are like a whole lifetime" [An-Nasaa’i]