Imam Muslim reports from Abu Huraira (RA), that the Prophet ﷺ said,
عَنْ أَبِي هُرَيْرَةَ، قَالَ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم ‏ “‏ أَفْضَلُ الصِّيَامِ بَعْدَ صِيَامِ شَهْرِ رَمَضَانَ شَهْرُ اللَّهِ الْمُحَرَّمُ ‏”
Abu Hurairah (RA) narrated that the Messenger of Allah ﷺ said: “The most virtuous fasting after the month of Ramadan is Allah’s month Al-Muharram.” [Jami` at-Tirmidhi 740]

This refers to general voluntary fasts according to Imam Ibn Rajab (Allah have mercy on him): These are best in the month of Muharram, just as the best general voluntary prayer is night prayer.

The virtue and honour of this month can be attested to by the fact that the Prophet ﷺ called it, “the Month of Allah.” Such ascription is only made by Allah to the most special of His creation, such as the ascription of the Prophets Muhammad, Ibrahim, Ishaq, Ya`qub, and others to his slave hood (Allah’s peace and blessings be on them all).

Given that Allah ascribed fasting, between all spiritual works, to Himself [saying, “It is Mine,”] it was suitable that this month, which is also ascribed to Allah, be selected for this particular form of worship.

Fasting is a secret between the servant and his Lord. This is why Allah Mighty and Exalted says, [in the divine hadith (hadith qudsi)], “Every action of the son of Adam is his, except for fasting. It is Mine, and it is I who rewards it.” [Bukhari and Muslim]
It is mentioned in Bukhari and Muslim from Ibn Abbas (RA) that he was asked about fasting the Day of `Ashura [10th of Muharram]. He said, “I did not see the Messenger of Allah ﷺ fast a day while more avid to seek its virtue than this day,” [meaning the Day of `Ashura]. [Bukhari and Muslim].

The Day of `Ashura has great virtue, and tremendous sanctity (hurma), which can be gauged from this beautiful Hadith.
عن أبي قتادة رضي الله عنه أن رسول الله صلى الله عليه وسلم سئل عن صيام يوم عاشوراء فقال‏:‏ ‏ “‏يكفر السنة الماضية‏”
‏Abu Qatadah (RA) reported: The Messenger of Allah ﷺ was asked about observing Saum (the fast) on the tenth day of Muharram, and he replied, “It is an expiation for the sins of the preceding year.” [Muslim].

When the Prophet of Allah ﷺ migrated to Madinah, and found the People of the Book fasting this day and venerating it, he ordered the Muslims to fast it, and encouraged it so much that even the children would fast it.

عَنْ أَبِي مُوسَى ـ رضى الله عنه ـ قَالَ دَخَلَ النَّبِيُّ صلى الله عليه وسلم الْمَدِينَةَ وَإِذَا أُنَاسٌ مِنَ الْيَهُودِ يُعَظِّمُونَ عَاشُورَاءَ وَيَصُومُونَهُ فَقَالَ النَّبِيُّ صلى الله عليه وسلم ‏ “‏ نَحْنُ أَحَقُّ بِصَوْمِهِ ‏”‏‏.‏ فَأَمَرَ بِصَوْمِهِ‏.‏
Narrated Abu Musa (RA): When the Prophet ﷺ arrived at Medina, he noticed that some people among the Jews used to respect Ashura’ (i.e. 10th of Muharram) and fast on it. The Prophet (ﷺ) then said, “We have more right to observe fast on this day.” and ordered that fasting should be observed on it.

At the end of his life, the Messenger of Allah ﷺ made the determination not to fast this day alone, but with another day [either before or after it], in order to be different from the People of the Book.

عَنِ ابْنِ عَبَّاسٍ، قَالَ: قَالَ رَسُولُ اللَّهِ ـ صلى الله عليه وسلم ـ ‏ “‏ لَئِنْ بَقِيتُ إِلَى قَابِلٍ لأَصُومَنَّ الْيَوْمَ التَّاسِعَ ‏”
It was narrated from Ibn ‘Abbas (RA) that the Messenger of Allah ﷺ said: “If I live until next year, I will fast the ninth day (of Muharram) too.” [Sunan Ibn Majah 1736]