As for fasting six days of Shawwāl, it is authentically reported from him that fasting them along with Ramadan is equivalent to fasting for a lifetime (1).
As for fasting (2) on the day of ‘Āshūrā’, he was more intent on fasting it than on other days. When he arrived in Medina, he found the Jews fasting on that day and holding it in honour. He said: “We have a greater claim to Moses than you do.” He therefore fasted it and instructed others to fast it (3). This was before fasting Ramadan became obligatory. Once it became obligatory, he said: “Whoever wishes may fast it, and whoever wishes may leave it” (4).
Some people (5) have found this problematic, saying: The Messenger of Allah (peace and blessings be upon him) arrived in Medina in the month of Rabī‘ al-Awwal. How, then, can Ibn ‘Abbās say that he arrived in Medina and found the Jews fasting on the day of ‘Āshūrā’?
There is another difficulty here: it is established in the two Sahīh collections (6) that ‘Ā’ishah said: “Quraysh used to fast on ‘Āshūrā’ during the pre-Islamic period, and the Messenger of Allah (peace and blessings be upon him) also used to fast it. When he emigrated to Medina, he fasted it and instructed others to fast it (7). When fasting the month of Ramadan became obligatory, he said: ‘Whoever wishes may fast it, and whoever wishes may leave it.’”