All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The Day of Ashura is the tenth day of Muharram. It is the day when Allah saved Musa (peace be upon him) and his people from Pharaoh and his armies, so Musa fasted it in gratitude to Allah. Our Prophet Muhammad (peace and blessings be upon him) also fasted it and recommended its observance. Ibn Abbas (may Allah be pleased with them) reported: "When the Prophet (peace be upon him) came to Medina, he saw the Jews fasting on Ashura. He asked: "What is this?" They replied: "This is a blessed day - the day Allah saved the Children of Israel from their enemy, so Musa fasted it." The Prophet said: "We have more right to Musa than you," so he fasted it and commanded its observance." [Al-Bukhari]
Fasting Ashura was established by our Prophet (peace be upon him) for this Ummah and he encouraged it. Ibn Abbas (may Allah be pleased with them) said: "I never saw the Prophet (peace be upon him) so keen to fast any day more than this day (Ashura) and this month (meaning Ramadan)." [Agreed upon]. Besides, The Prophet (peace be upon him) said: "Fasting Arafah expiates two years (the previous and coming), and fasting Ashura expiates one year." [An-Nisa'i /Al-Sunn Al-Kubra]
For a Muslim to attain the greatest reward from Allah, it is best to fast three days: the ninth, tenth and eleventh of Muharram, as mentioned by many scholars including As-Shafi'i. If one cannot do this, they should fast either the day before or after Ashura. If one fasts only Ashura, that is permissible, and all these options are good. In this regard, Al-Imam An-Nawawi (may Allah have mercy on him) said: "Ashura is the tenth of Muharram, and it is recommended to fast Tasu'a (the ninth) with it." [Rawdat At-Talibin]
Thus, fasting Ashura has three levels:
1. The minimum: fasting only Ashura
2. Better: fasting the ninth with it
3. Best: fasting the ninth, tenth and eleventh
Fasting Ashura remains recommended even if it falls on a Saturday, and there is no prohibition in doing so, as there is a specific reason for fasting. The Messenger of Allah (peace be upon him) said: "The best fast is the fast of my brother Dawud; he would fast one day and not fast the next." [At-Tirmidhi] This necessarily includes fasting on Saturdays, and this is the consensus of the scholars of the accepted schools.
Furthermore, Al-Imam Ar-Ramli (may Allah have mercy on him) said: "It is disliked to single out Saturday for fasting...unless it coincides with one's regular fasting habit, such as fasting alternate days, or fasting Ashura or Arafah - then there is no dislike." [Nihayat Al-Muhtaj/Vol. 3/P.209]. And Allah The Almighty Knows Best.