Fasting is required in Ramaḍān but there are other fasts outside of that month which are recommended as well.
The month following Ramaḍān is known as Shawwāl and it is highly recommended to fast any six days of that month. They do not have to be consecutive. The Prophet said, “Whoever fasted in Ramaḍān and then followed it up with six days of Shawwāl, it is as if he fasted continuously.” That is because the Prophet taught that good deeds are rewarded at least ten times, so fasting Ramaḍān is like fasting ten months. By fasting six more days, it is like fasting sixty more days, making a total of 12 months, so it is as if the person has fasted the entire year.
The fasts of Shawwāl do not have to be done in order and may be performed on any day of the month. A common question about fasting in Shawwāl is whether a person who has to make up missed fasts can combine the intention of making up a fast and also having it count as one of the six fasts in Shawwāl. This should not be done because the one who has missed fasts from Ramaḍān has not completed the month of fasting, so the purpose of earning the reward for fasting the entire year cannot be multiplied until that is completed separately. Either one can be done first.
It is also highly recommended to fast on the tenth of Muḥarram [known as the day of ʿĀshūrā’] and on the first nine days of Dhul Ḥijjah, with the ninth being more emphasized than the other eight days.
If one would like to fast more regularly throughout the year then it is recommended to fast three days, preferably the 13th, 14th, and 15th of any given month [of the lunar calendar]. These are known as the ‘white days’ because the moon is at its largest size. It is also recommended to fast every Monday or Thursday, or both, because that was the habit of the Prophet.
One may fast on almost any day as an act of worship. However there are some days on which it is disliked to fast and others on which it is prohibited. It is disliked to fast on a Friday or Saturday unless it was done for a specific reason, like if it happened to be the day of ʿĀshūrā’, for example. It is prohibited to fast on the day of Eid al-Fiṭr, Eid al-Aḍḥā, and on the 11th, 12th, and 13th of Dhul Ḥijjah. It is also prohibited to fast every day with the intention to continue throughout your life because that would weaken the body and make you accustomed to fasting, so it would lose its purpose. It is also prohibited to fast for two days straight without breaking the fast in between.
 Muslim #1164