Six Fasts of Shawwal

I do not know Allah's Messenger (ﷺ) singling out any day's fast and considering it more excellent than another, except this day (the day of Ashura) and this month, meaning the month of Ramadan.

I asked or someone asked the Prophet (ﷺ) about perpetual fasting. He replied: You have a duty to your family. Fast during Ramadan and the following month, and every Wednesday and Thursday. You will then have observed a perpetual fast.

the Messenger of Allah said: "Whoever fasts Ramadan, then follows it with six from Shawwal, then that is (equal in reward) to fasting everyday."

"My paternal uncles among the Ansar who were among the Companions of the Messenger of Allah (ﷺ) told me: "The new crescent of Shawwal was covered with clouods, so we fasted the next day. Then some riders came at the end of the day and testified to the Prophet (ﷺ) that they had seen the new crescent the night before. The Messenger of Allah (ﷺ) commanded them to break their fast and to go out to offer the "Eid prayer the following morning.'"

"When you see the new crescent, fast, and when you see it, stop fasting. If it is cloudy then calculate it (as thirty days)." Ibn "Umar used to fast one day before the new crescent was seen.

the Prophet (ﷺ) forbade every wild animal that has fangs and every bird that has talons, the price of dead meat, the flesh of donkeys, the wages of a prostitute, the stud fee of the stallion and purple saddle cloths.

On one occasion all the wives of the Prophet (ﷺ) were with him when his daughter, Fatimah (May Allah be pleased with her) who walked after the style of his father, came there. He (ﷺ) welcomed her saying, "Welcome, O my daughter", and made her sit on his right side, or on his left side and then whispered something to her at which she wept bitterly. When he perceived her grief, he talked secretly to her again and she smiled (with happiness). I said to her: "Messenger of Allah (ﷺ) chose you from amongst all his wives to speak secretly to you and yet you cried." When he left, I asked her, "What did Messenger of Allah (ﷺ) say to you?" She said, "I will not divulge the secret of Messenger of Allah (ﷺ)." When Messenger of Allah (ﷺ) passed away, I said to her: "I adjure you by the right I have in respect of you to tell me what Messenger of Allah (ﷺ) had told you." She said: "Now (when Messenger of Allah (ﷺ) has died), I will tell you. When he whispered to me the first time, he told me, 'Jibril (Gabriel) used to listen to my recitation of the Qur'an and then recite it back to me once or twice a year, and this time he has done it twice; and so I perceive that my death is approaching. Then be mindful of your duty to Allah and be patient and steadfast, for I shall be an excellent predecessor for you.' On this I wept as you saw. When he perceived my distress he talked to me secretly the second time and said, 'O Fatimah, are you not pleased that you will be the chief among the believing women or of this Ummah?' This made me smile as you saw."

[Al-Bukhari and Muslim].