Shaykh Ibn ‘Uthaymeen said:
The fast of ‘Aashooraa has four levels.
The first level is to fast the 9th, the 10th and the 11th [days of Muharram].
And this is the highest level, based upon what [Imaam] Ahmad transmitted in al-Musnad:
«Fast the day before it, and the day after it; [and in doing so] differ [yourselves] from the Jews.»
And [in addition], if a person was to fast these three days, then he has attained the virtue of fasting the [recommended] three days of every month.
The second level, [is to fast] the 9th and the 10th [days of Muharram], based on the statement of the Prophet ﷺ:
«If I live until next year, I will certainly fast the ninth [day of Muharram].»
…[and] he said this after it was said to him that the Jews used to fast the 10th day [of Muharram]; and he used to like to differ from the Jews, rather to differ from every disbeliever.
And the third [level], [is to fast] the 10th and the 11th [days of Muharram].
And the fourth [level], [is to fast] the 10th [day of Muharram] by itself.
And from amongst the scholars are those who say [it] is permissible [to do so], and from amongst them are those who say [it] is disliked [to do so].