The month of Muharram is the first month of the Arabic year, and it is one of the four sacred months of Allah. Allah says in the Quran:
“Verily, the number of months with Allah is twelve months (in a year), so was it ordained by Allah on the Day when He created the heavens and the earth; of them four are Sacred (i.e. the 1st, the 7th, the 11th and the 12th months of the Islamic calendar)….” [Quran, Surah Al-Tawbah 9:36]
The Prophet (peace and blessings of Allah be upon him) said: “The division of time has turned to its original form which was current the day Allah created the heavens and earth. The year consists of twelve months of which four are sacred: three consecutive months, Dhu’l-Qa’dah, Dhu’l-Hijjah and Muharram, and Rajab of Mudar which comes between Jumada and Sha’baan.” [Hadith mentioned in al-Bukhaari (3167) and Muslim (1679) as narrated from Abu Bakrah (may Allah be pleased with him)]
The Messenger of Allah (peace and blessings of Allah be upon him) said: “The best fasting after Ramadan is the month of Allah Muharram, and the best prayer after the obligatory prayer is prayer at night.” Narrated by Muslim, 1163.
It was proven that the Prophet (peace and blessings of Allah be upon him) did not fast any month in full apart from Ramadan, so this hadith is to be understood as meaning that it is it is encouraged to fast a lot in the month of Muharram, not to fast the whole month.