The scholars regarded it as mustahabb to fast on the eleventh of Muharram because it was narrated from the Prophet (blessings and peace of Allah be upon him) that he instructed people to fast on this day. This is seen in the report narrated by Ahmad from Ibn ‘Abbas (may Allah be pleased with him) who said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Fast the day of 'Ashura, and be different from the Jews with regard to it; fast the day before it or the day after it.”
The scholars differed regarding the soundness of this hadith. Shaykh Ahmad Shaakir classed it as hasan, but the editors of the Musnad classed it as da‘eef.
It was also narrated by Ibn Khuzaymah with the same wording. Al-Albaani said: Its isnaad is da‘eef, because of the poor memory of Ibn Abi Layla, and it was contradicted by a mawqoof report narrated by ‘Ata’ and others from Ibn ‘Abbas, the isnaad of which is saheeh according to at-Tahhaawi and al-Bayhaqi. End quote.
If the hadith is hasan, that is all well and good. If it is da‘eef, then da‘eef hadiths about such matters are tolerated by the scholars, because its weakness is minor; it is not false or fabricated, and it speaks about virtuous deeds. Moreover, it is narrated from the Prophet (blessings and peace of Allah be upon him) that he encouraged fasting in the month of Muharram. Indeed the Messenger of Allah (blessings and peace of Allah be upon him) said: “The best fast after Ramadaan is the month of Allah Muharram.” Narrated by Muslim, 1163.
Al-Bayhaqi narrated this hadith in as-Sunan al-Kubra with the wording mentioned above, and in another report with the wording “Fast one day before it and one day after it,” saying “and” instead of “or”.
Al-Haafiz Ibn Hajar narrated it in It-haaf al-Mahrah (2225) as follows: “Fast one day before it and one day after,” and he said: It was narrated by Ahmad and al-Bayhaqi with a da‘eef isnaad, because of the weakness of Muhammad ibn Abi Layla, but he is not the only one who narrated it; Saalih ibn Abi Saalih ibn Hayy followed him in that. End quote.
This report indicates that it is mustahabb to fast on the ninth, the tenth and the eleventh.
The scholars also mentioned another reason for it being mustahabb to fast on the eleventh, which is to be on the safe side with regard to the tenth day, because people may be mistaken in sighting the new moon of Muharram, and it may not be known exactly which day is the tenth. So if the Muslim fasts on the ninth, the tenth and the eleventh, then he will have fasted 'Ashura for certain. Ibn Abi Shaybah narrated in al-Musannaf (2/313) from Tawoos (may Allah have mercy on him) that he used to fast one day before it and one day after it, for fear of missing it.
Imam Ahmad said: whoever wants to fast 'Ashura should fast on the ninth and the tenth, unless there is some uncertainty about the months, in which case he should fast for three days. Ibn Sireen said that. Al-Mughni, 4/441
Thus it becomes clear that it is not correct to describe fasting three days as being an innovation (bid‘ah).
But if a person misses fasting on the ninth, and he fasts only on the tenth, there is nothing wrong with that, and it is not makrooh. But if he adds the eleventh to it, that is better. Al-Mardaawi said in al-Insaaf (3/346):
It is not makrooh to single out the tenth for fasting, according to the correct view. Shaykh Taqiy ad-Deen (Ibn Taymiyah) agreed that it is not makrooh. End quote.
And Allah knows best.