Short Answer
It is recommended to fast six days of Shawwāl, to fast three days (preferably the 13th, 14th, and 15th) of any given month (of the lunar calendar), to fast every Monday or Thursday (or both), to fast on the 10th of Muḥarram (known as the day of ʿĀshūrā’), and to fast on the first nine days of Dhul Ḥijjah (with the 9th being more emphasized). Do not fast on the day of Eid al-Fiṭr, Eid al-Aḍḥā, and on the 11th, 12th, and 13th of Dhul Ḥijjah. Also, do not fast every day with the intention to continue throughout your life or to fast for two days straight without breaking the fast in between.
What the Prophet Said
“Whoever fasted Ramaḍān and then followed it up with six days of Shawwāl, it is as if he fasted continuously.” [Muslim #1164]
“Fast some days of the sacred months and not others.” [Abu Dawud, #2428 – graded weak by several scholars]
What the Scholars Said
Imam Ibn Taymiyyah: “As for fasting in Rajab specifically, the hadiths about that are all weak, actually they are fabricated. The scholars do not rely on any of them. They are not among the weak hadiths which have been narrated about virtues, rather most of them are fabricated and false. In Al-Musnad and elsewhere there is a hadith which says that the Prophet encouraged fasting the sacred months, namely Rajab, Dhu’l-Qa’dah, Dhu’l-Hijjah and Muharram, but this has to do with fasting during all of them, not just Rajab.” [Majmu’ Al-Fatawa, 25:290]
Imam Ibn al-Qayyim: “Every hadith which mentions fasting in Rajab and praying during some of its nights is false and fabricated.” [Al-Manar al-Munif, 96]
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Shaykh Mustafa Umar