The Prophet (SAW) observed fasting on the day of A’shura and ordered his companions to fast as well. There are many authentic Hadiths to this effect.
According to a narration reported by A’ishah (Allah be pleased with her)
(The tribe of) Quraish used to fast on the day of A’shura in the Pre-Islamic period, and then Allah’s Apostle ordered (Muslims) to fast on it till the fasting in the month of Ramadan was prescribed; whereupon the Prophet (ﷺ) said, “He who wants to fast (on ‘A’shura’) may fast, and he who does not want to fast may not fast.” (Sahih al-Bukhari- 1893 & Sahih Muslim – 2499)
Ibn Abbas (Allah be pleased with both of them) reported that when Allah’s Messenger (may peace be upon him) came to Medina, he found the Jews observing the fast on the day of A’shura. They (the Jews) were asked about it and they said: It is the day on which Allah granted victory to Moses and (his people) Bani Isra’il over the Pharaoh and we observe fast out of gratitude to Him. Upon this the Apostle of Allah (may peace be upon him) said: We have a closer connection with Moses than you have, and he commanded to observe fast on this day. (Sahih Muslim – 2520)
‘Ubaydallah ibn Abi Yazid reported: Ibn Abbas, may Allah be pleased with him, said, “I did not see the Prophet, peace and blessings be upon him, as eager to fast a day and preferring it over others except for this day of Ashura and this month of Ramadan.”(Ṣaḥīḥ al-Bukhārī 1902, Ṣaḥīḥ Muslim 1132)
Fasting on the A’shura day is a Sunnah, strongly recommended ritual. Imam Al-Nawawi said, “Al-Shafi’i and his companions, Ahmad, Ishaq, and others said it is recommended to fast the ninth and tenth of al-Muharram together, as the Prophet (ṣ) fasted the tenth and intended to fast the ninth.”
Ibn Hajar said, “The fast of A’shura has three levels: the lowest is to fast one day, the next is to fast the ninth day along with it, and the next is to fast the ninth, tenth, and eleventh days. Allah knows best.”(Fatḥ al-Bārī 1896
The Prophet was asked about fasting the day of A’shura and he said, “It will expiate the sins of the past year.” (Ṣaḥīḥ Muslim 1162)