Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
It is recommendable to fast as much of the month of Muharram as one can; particularly the 9th, the 10th and the 11th days of it. Added to these are the established recommendation of fasting the 13th, 14th and the 15th of each lunar month.
Answering your question, Dr. Muhammad Salama, PhD, Islamic Studies in English,Associate Professor at the Faculty of Islamic Sciences al-Madinah International University (Mediu), states:
Fasting in the month of Muharram is highly recommendable. The Messenger (peace and blessings be upon him) said, “The best month for observing Fasting after the month of Ramadan is the month of Muharram.” (Muslim)
The tenth day of that month, known as `Ashura’, is most recommended to fast on as the Messenger of Allah (peace and blessings be upon him) told that such fasting causes minor sins to be forgiven for the year that went before.
Fasting that day was even obligatory before the obligation of fasting Ramadan, but then it became only recommendable; though the Messenger of Allah used to observe fasting on it.
Ibn `Abbas said “I never saw the Prophet (peace and blessings be upon him) seeking to fast on a day more preferable to him than this day, the day of `Ashura’…”
In another authentic hadith, the Prophet recommended fasting on the 9th day of Muharram along with the 10th to distinguish Muslim’s fasting from the one observed by the Jewish on the 10th day. In a less authentic narration, it is also recommendable to fast on the 11th day for the same reason.
To sum up, it is recommendable to fast as much of that month as one can; particularly the 9th, the 10th and the 11th days of it. Added to these are the established recommendation of fasting the 13th, 14th and the 15th of each lunar month.
Allah Almighty knows best.