Prophet Muhammad (pbuh) had ordered Muslims to fast on Ashura days (9th, 10th, 11th days of Muharram) before the verses about Ramadan fasting were sent down. After the verses were sent down, he made it optional to fast or not. So, there is no special obligatory worship to perform in the month of Muharram now.
Narrations about Ashura Day Fasting are listed below:
1- A’isha, the wife of the Prophet, said, “The day of Ashura was a day the Quraysh used to fast in the Jahiliyya, and the Messenger of Allah also used to fast it during the Jahiliyya. When the Messenger of Allah came to Madina he fasted it and ordered that it be fasted. Then Ramadan fasting was made obligatory and that became the duty (fard) instead of Ashura. After that, whoever preferred fasted the Ashura, whoever not, did not fast.
(Bukhari, Sawm, 1, 68; Muslim, Siam, 113 (1125); Tirmidhi, Sawm, 49; Abu Dawood, Sawm 64; Muwatta, Siam, 33)
2- Narrated Ibn ‘Umar: Fasting was observed on the day of Ashura (10th of Muharram) by the people of the Pre-lslamic Period. But when (the order of compulsory fasting in) the month of Ramadan was revealed, the Prophet said, “It is up to one to fast on it (Ashura) or not.”
(Bukhari, Sawm, 68; Muslim, Siam, 117 (1126); Abu Dawood, Sawm, 64)
3- Narrated Ibn ‘Abbas: When the Prophet arrived at Medina he found that the Jews observed fast on the day of Ashura. They were asked the reason for the fast. They replied, “This is the day when Allah caused Moses and the children of Israel to have victory over Pharaoh, so we fast on this day as a sign of glorifying it.” The Prophet said, “We are closer to Moses than you.” Then he ordered fasting on this day to Muslims.
(Buxhari, Anbiya, 24, Manakibu’l-Ansar, 52; Muslim, Siam, 127 (1130); Abu Dawood, Sawm, 64; Ibn Majah, Siam, 41; Ahmad ibn Hanbal, 1/291, 310, 336, 340)
The first 2 narrations express that the Prophet fasted in Mecca on Ashura. However it is understood from the last narration that this fasting has actually started as an obligatory duty in Madinah and continued to be so until the verses about Ramadan fasting were revealed.
Currently, fasting on the Day of Ashura is yet another optional (nafilah) fast.