It is Mustahab (act of great merit) to observe the six fasts of Shawwaal. The observance of these fasts has been stressed by Rasulullah (sallallahu alayhi wasallam) in several ahadith. It is once again the infinite mercy of Allah Ta’ala, that in exchange of minimal efforts he accords multiple rewards. This is a clear indication that Allah Ta’ala seeks the slightest opportunity to draw his bondsmen closer to him.
We learn from Hadith that a person who observes these fasts after observing the fasts of Ramadaan receives the reward of fasting for the entire year.
Hadhrat Abu Ayyub (radiyallahu anhu) narrates that Rasulullah (sallallahu alayhi wasallam) said, “Whoever fasted the full month of Ramadaan and then follows it with six fasts of Shawwaal, is like a person who has fasted the entire year.” (Targhib)
Hazrat Ibn Umar (radiyallahu anhuma) narrates that Rasulullah (sallallahu alayhi wasallam) said, “One who fasted during the month of Ramadaan and further fasted six days in Shawwaal, is cleansed from sins like the day he was born.” (Targhib)
These fasts may be kept consecutively or at intervals during the month of Shawwaal