1. Ibn ‘Abbas (RadiAllahu anhuma) reported that when the Messenger of Allah (Sallellahu alaihi wa sallam) fasted on the day of ‘Ashura and commanded that it should be observed as a fast, they (his Companions) said to him: Messenger of Allah, it is a day which the Jews and Christians hold in high esteem. Thereupon the Messenger of Allah (Sallellahu alaihi wa sallam) said: When the next year comes, God willing, we would observe fast on the 9th But the Messenger of Allah (sallellahu alaihi wa sallam) died before the advent of the next year.(So we muslims have fasting on 9 and 10 both)
2. Ibn’Abbas (r.a) reported that the Messenger of Allah (s.a.w.s.) arrived in Medina and found the Jews observing fast on the day of ‘Ashura. The Messenger of Allah (s.a.w.s.) said to them: What is the (significance) of this day that you observe fast on it? They said: It is the day of great (significance) when Allah delivered Moses and his people, and drowned the Pharaoh and his people, So we observe fast out of gratitude . Upon this the Messenger of Allah (s.a.w.s.) said: We have more right, and we have a closer connection with Moses than you have; so Allah’s Messenger (s.a.w.s.) observed fast (on the day of ‘Ashura), and gave orders that it should be observed.
3. Abu Haraira (r.a.) reported Allah’s Messenger (s.a.w.s.) as saying: The most excellent fast after Ramadan is God’s month. al-Muharram, and the most excellent prayer after what is prescribed is prayer during the night.
4. Abu Qatada al-Ansari (r.a.) reported that the Messenger of Allah (s.a.w.s.) was asked about was asked about fasting on the day of ‘Ashura (10th of Muharram), whereupon be said: It expiates the sins of the preceding year.
[Sahih Muslim, The Book of Fasting (Kitab Al-Sawm)]