Abu Hurairah (radiyallahu ta’ala ‘anhu) reported: The Messenger of Allah (salallahu ‘alahi wasallam) said, “The best month for observing Saum (fasting) next after Ramadan is the month of Allah, the Muharram; and the best salat next after the prescribed salat is salat at night (Tahajjud prayers).”
(Muslim)

Ibn ‘Abbas (radiyallahu ta’ala ‘anhu) reported that the Prophet (salallahu ‘alahi wasallam) came to Madinah and saw the Jews fasting on the day of ‘Ashura. He asked: “What is this?” They said: “This is a righteous day, it is the day when Allah saved the Children of Israel from their enemies, so Musa fasted on this day.” He said: “We have more right to Musa than you,” so he fasted on that day and commanded [the Muslims] to fast on that day. (Bukhari)

Ibn Abbas (radiyallahu ta’ala ‘anhu) reported: The Messenger of Allah (salallahu ‘alahi wasallam) observed Saum (fasting) on the day of Ashura and commanded us to fast on this day.
(Bukhari, Muslim)

Abu Qatadah (radiyallahu ta’ala ‘anhu) reported: The Messenger of Allah (salallahu ‘alahi wasallam) was asked about observing As-Saum (the fast) on the tenth day of Muharram, and he replied, “It is an expiation for the sins of the preceding year.”
(Muslims)