Imam Mansur ibn Yunus Al-Buhuti (d. 1051 AH), may Allah be pleased with him, said of fasting on 10th Muharram:
So it is highly emphasised to fast the 10th of Muharram as well as the 9th day beforehand. This is based upon the hadith of the Prophet, peace and blessings of Allah be upon him, “…Let it be the case that you fast the ninth and then the tenth of Muharram.”
Taken from Ar-Rawd ul-Murbi
bi-Sharhi Zad il-Mustaqni, pp. 292-293, Dar ul-Khair, Beirut 1423 AH
Imam Ibrahim ibn Muhammad ibn Salim ibn Duwayan (d. 1353 AH), may Allah have mercy upon him, said the following regarding fasting on `Ashura’, known as 10th of Muharram:
It is the sunnah to fast in the month of Muharram based upon the hadith of Abu Hurairah, may Allah be pleased with him, that is attributable, “The best fast after that of Ramadan is Muharram, the Month of Allah.” This is collected by Imam Muslim [vol.2, pp. 820-821].
The most emphasised fast of this month is Ashura’, which is an expiation for an entire year. This understanding comes from the hadith of Abu Qatadah in which the Prophet, peace and blessings of Allah be upon him, said regarding fasting the Day of `Ashura, “The one that fasts on that day shall have Allah expiate from him the sins of the following year.” This is collected by Imam Muslim [vol.2, pp. 818-819].
Taken from Manar us-Sabil fi Sharh id-Dalil, pp. 195-196, Dar Ibn Hazm edition, Beirut 1426 AH