In the name of Allah, the Gracious, the Merciful
Al-Nawawi writes in his book Minhaj al-Talibin:
يسن صوم الاثنين والخميس وعرفة وعاشوراء وتاسوعاء وأيام البيض وستة من شوال وتتابعها أفضل ويكره إفراد الجمعة وإفراد السبت
It is the Sunnah to fast on Monday and Thursday, the day of Arafat, the day of Ashura and the ninth of al-Muharram, the white days (the 13th, 14th, and 15th of each month), and six days of Shawwal and to do so consecutively is better. It is disliked to single out Friday or Saturday for fasting.
Source: Minhāj al-Ṭālibīn 1/79
Al-Nawawi recorded in his book Riyad al-Salihin:
Abu Huraira reported: The Messenger of Allah, peace and blessings be upon him, said:
أَفْضَلُ الصَّلاَةِ بَعْدَ الصَّلاَةِ الْمَكْتُوبَةِ الصَّلاَةُ فِي جَوْفِ اللَّيْلِ وَأَفْضَلُ الصِّيَامِ بَعْدَ شَهْرِ رَمَضَانَ صِيَامُ شَهْرِ اللَّهِ الْمُحَرَّمِ
The best prayer after the prescribed prayers is a prayer offered in the middle of the night. The best fast after the month of Ramadan is a fast in the month of Allah, al-Muharram.
Source: Ṣaḥīḥ Muslim 1163, Grade: Sahih
Aisha reported:
مَا رَأَيْتُ رَسُولَ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ اسْتَكْمَلَ صِيَامَ شَهْرٍ إِلَّا رَمَضَانَ وَمَا رَأَيْتُهُ أَكْثَرَ صِيَامًا مِنْهُ فِي شَعْبَانَ
I did not see the Messenger of Allah, peace and blessings be upon him, complete an entire month of fasting except for Ramadan, and I did not see him complete more voluntary fasts than in the month of Sha’ban.
Source: Ṣaḥīḥ al-Bukhārī 1969, Muttafaqun Alayhi
In another narration, Aisha said:
كَانَ يَصُومُ شَعْبَانَ إِلاَّ قَلِيلاً
The Prophet would fast all of Sha’ban except for a few days.
Source: Ṣaḥīḥ Muslim 1156, Grade: Sahih
Ibn Abbas reported: The Prophet, peace and blessings be upon him, said:
مَا الْعَمَلُ فِي أَيَّامٍ أَفْضَلَ مِنْهَا فِي هَذِهِ
No good deeds are better than what is done in these first ten days of Dhul Hijjah.
Some companions of the Prophet said, “Not even jihad in the way of Allah?” The Prophet said:
وَلَا الْجِهَادُ إِلَّا رَجُلٌ خَرَجَ يُخَاطِرُ بِنَفْسِهِ وَمَالِهِ فَلَمْ يَرْجِعْ بِشَيْءٍ
Not even jihad in the way of Allah, except for a man who goes out with his life and wealth at risk and he returns with nothing.
Source: Ṣaḥīḥ al-Bukhārī 969, Grade: Sahih
Abu Qatadah reported: The Prophet, peace and blessings be upon him, was asked about fasting the day of Arafat. The Prophet said:
يُكَفِّرُ السَّنَةَ الْمَاضِيَةَ وَالْبَاقِيَةَ
It will expiate the sins of the previous and upcoming years.
Then he was asked about fasting the day of Ashura. The Prophet said:
يُكَفِّرُ السَّنَةَ الْمَاضِيَةَ
It will expiate the sins of the year.
Source: Ṣaḥīḥ Muslim 1162, Grade: Sahih
Ibn Abbas reported: The Messenger of Allah, peace and blessings be upon him, said:
لَئِنْ بَقِيتُ إِلَى قَابِلٍ لَأَصُومَنَّ التَّاسِعَ يَعْنِي يَوْمَ عَاشُورَاءَ
If I live until next year, I will certainly fast the ninth day of Al-Muharram before Ashura.
Source: Ṣaḥīḥ Muslim 1134, Grade: Sahih
Abu Ayyub reported: The Messenger of Allah, peace and blessings be upon him, said:
مَنْ صَامَ رَمَضَانَ ثُمَّ أَتْبَعَهُ سِتًّا مِنْ شَوَّالٍ كَانَ كَصِيَامِ الدَّهْرِ
Whoever fasts the month of Ramadan and then follows it with six days of fasting during the month of Shawwal will be as if he fasted for the year.
Source: Ṣaḥīḥ Muslim 1164, Grade: Sahih
Abu Qatadah reported: The Messenger of Allah, peace and blessings be upon him, was asked about fasting on Monday. The Prophet said:
ذَاكَ يَوْمٌ وُلِدْتُ فِيهِ وَيَوْمٌ بُعِثْتُ أَوْ أُنْزِلَ عَلَىَّ فِيهِ صَوْمُ ثَلاَثَةٍ مِنْ كُلِّ شَهْرٍ وَرَمَضَانَ إِلَى رَمَضَانَ صَوْمُ الدَّهْرِ
That is the day I was born and the day I was sent with revelation. Fasting three days of every month and Ramadan is like fasting for a lifetime.
Source: Ṣaḥīḥ Muslim 1162, Grade: Sahih
Abu Huraira reported: The Messenger of Allah, peace and blessings be upon him, said:
تُعْرَضُ الْأَعْمَالُ يَوْمَ الِاثْنَيْنِ وَالْخَمِيسِ فَأُحِبُّ أَنْ يُعْرَضَ عَمَلِي وَأَنَا صَائِمٌ
The deeds are presented on Monday and Thursday. Thus, I love for my deeds to be presented while I am fasting.
Source: Sunan al-Tirmidhī 747, Grade: Sahih
Aisha reported:
كَانَ النَّبِيُّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ يَتَحَرَّى صَوْمَ الِاثْنَيْنِ وَالْخَمِيسِ
The Prophet, peace and blessings be upon him, was eager to fast on Monday and Thursday.
Source: Sunan al-Tirmidhī 745, Grade: Sahih
Abu Huraira reported:
أَوْصَانِي خَلِيلِي (النبي صلى الله عليه وسلم) بِثَلَاثٍ لَا أَدَعُهُنَّ حَتَّى أَمُوتَ صَوْمِ ثَلَاثَةِ أَيَّامٍ مِنْ كُلِّ شَهْرٍ وَصَلَاةِ الضُّحَى وَنَوْمٍ عَلَى وِتْرٍ
My dear friend the Prophet, peace and blessings be upon him, advised me to do three deeds and I will never abandon them until I die: to fast three days of every month, to perform forenoon prayer, and to sleep after performing the witr prayer.
Source: Ṣaḥīḥ al-Bukhārī 1178, Grade: Muttafaqun Alayhi
Success comes from Allah, and Allah knows best.