Recommended fasts include Mondays/Thursdays, the three white days each month, six days of Shawwal, and عرفة/Ashura days — avoid fasting on Eid and التشريق days.
Voluntary fasting after الحج is an excellent way to continue building on the spiritual foundation أُسس during the الحج. The most significant post-الحج fasting opportunity is the six days of Shawwal — النبي (صلى الله عليه وسلم) said: 'Whoever fasts رمضان then follows it with six days of Shawwal, it is as if they fasted the entire year' (Muslim). Since الحج occurs in ذو الحجة (after Shawwal), this applies if you fast the following Shawwal.
Year-round مستحب fasts include: Mondays and Thursdays (النبي's regular practice, as these are days when deeds are presented to Allah), the three white days of each lunar month (13th, 14th, and 15th — Ayyam al-Beed), the Day of عرفة (9th ذو الحجة, for non-الحجاج), and the Day of Ashura (10th Muharram, with the 9th or 11th). Important: fasting is prohibited on the two Eid days and the three days of التشريق (11th-13th ذو الحجة), so new الحجاج should not attempt to fast immediately after their الحج rituals conclude.
Source: صحيح مسلم; سنن الترمذي; الإمام النووي, Riyad al-Salihin