Fasting the day of ‘Ashoora’ is expiation for one year,

the Prophet (peace and blessings of Allaah be upon him) said: “The best fasting after the month of Ramadaan is fasting in the month of Muharram.” (Narrated by Muslim, 1163)
=======
Shaykh al-Islam [Ibn Taymiyah] said: Fasting the day of ‘Ashoora’ is expiation for one year, and it is not makrooh to single out this day for fasting. Al-Fataawa al-Kubra, part 5.

In Tuhfat al-Muhtaaj by Ibn Hajar al-Haythami it says: There is nothing wrong with fasting ‘Ashoora’ on its own. Part 3, Baab Sawm al-Tatawwu’.

The Standing Committee was asked this question and replied as follows:

It is permissible to fast the day of ‘Ashoora’ on its own, but it is better to fast the day before it or the day after it. This is the Sunnah that is proven from the Prophet (peace and blessings of Allaah be upon him), as he said, “If I am still alive next year, I will certainly fast the ninth.” Narrated by Muslim, 1134.

Ibn ‘Abbaas (may Allaah be pleased with him) said: He meant, along with the tenth.

And Allaah is the Source of strength.

Al-Lajnah al-Daa’imah li’l-Buhooth al-‘Ilmiyyah wa’l-Ifta’, 11/401.