What are the most preferable days for fasting besides Ramadan?
All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
According to many Prophetic narrations, it is recommended for a Muslim to fast during the following days.
1) Six days of the month of Shawwaal.
Abu Ayyoob Al-Ansaari may Allah be pleased with him, relates that Allah's Messenger said: "Whoever fasts the month of Ramadan and then follows it with six days of Shawwaal will be rewarded as the one who fasts the entire year." [Muslim, Ahmad, Abu Daawood and At-Tirmithi]
2) The first nine days of Thul-Hijjah.
Hafsah may Allah be pleased with her, relates: "There are four things that the Prophet never abandoned: fasting the day of 'Aashooraa', fasting the first ten days of Thul-Hijjah, fasting three days of every month and praying the two Raka's before the dawn prayer." [Ahmed and An-Nasaa'ee] Note that "the first ten" means the first nine days since the tenth will be the 'Eed (i.e. the day of sacrifice).
3) The day of 'Arafah for those who are not performing Hajj.
Abu Qataadah, may Allah be pleased with him, relates that the Messenger of Allah said: "Fasting on the day of 'Arafah is an expiation for two years, the year preceding it and the year following it. The fasting of 'Aashooraa' is an expiation for the year preceding it." [Muslim, Abu Daawood, and Ibn Maajah] Abu Hurayrah, may Allah be pleased with him, reported: Allah's Messenger forbade fasting on the day of 'Arafah for one who is actually in 'Arafah." [Ahamad and Ibn Maajah]
4) Fasting during the month of Al-Muharram.
Abu Hurayrah may Allah be pleased with him, reported that when the Prophet was asked: "Which fast is better after Ramadan?' He said: "The month of Allah, Al-Muharram." [Muslim and others]
Also, fasting the day of 'Aashooraa' (10th of Muharram) and the day before it according to a narration from Ibn 'Abbaas, may Allah be pleased with him: "The Messenger of Allah fasted on the day of 'Aashooraa' and ordered the people to fast on it'. The people said: 'O Allah's Messenger! It is a day that the Jews and Christians honor. He said: 'When the following year comes - Allah willing - we shall fast on the ninth as well." [Ahmad and Muslim] The death of Allah's Messenger came before the following year.
5) Fasting most of the month of Sha'baan.
Aa'ishah may Allah be pleased with her, said: "I have never seen the Messenger of Allah fast more in a month than he did in Sha'baan." [Al-Bukhari and Muslim]
6) Fasting three days of every lunar month.
Abu Hurayrah and Abu Tharr, may Allah be pleased with them, replied: "The Messenger recommended us to perform the two Raka's of Adh-Dhuhaa, to perform al-Witr prayer before sleeping, and fast for three days of every month." [Muslim]
It is better to fast 13th, 14th and 15th days. Aa'ishah, may Allah be pleased with her, said: "The Prophet used to fast on Monday and Thursday." [Abu Daawood and others]
Finally, know that the days of optional fasting are many, so if one regularly fasts some of them it will be good and better than fasting for many days and then leave fasting. Imaam Muslim reported: "It is the Prophet's practice to perform his good deeds regularly and persistently."
Allah Knows best.