Hadith about the Month of Shawwal: Sahih Muslim - Hadith # 1423 a, 1164 a (English/Arabic)

The Book of Marriage

(11) Chapter: It is recommended to get married and arrange marriages in Shawwal, and it is recommended to consummate the marriage in that month

'A'isha (Allah be pleased with her) reported:

Allah's Messenger (ﷺ) contracted marriage with me in Shawwal and took me to his house as a bride during Shawwal. And who among the wives of Allah's Messenger (ﷺ) was dearer to him than I, and' A'isha liked that the women (of her family) should enter the houses as brides during the month of Shawwal.

حَدَّثَنَا أَبُو بَكْرِ بْنُ أَبِي شَيْبَةَ، وَزُهَيْرُ بْنُ حَرْبٍ، - وَاللَّفْظُ لِزُهَيْرٍ - قَالاَ حَدَّثَنَا وَكِيعٌ، حَدَّثَنَا سُفْيَانُ، عَنْ إِسْمَاعِيلَ بْنِ أُمَيَّةَ، عَنْ عَبْدِ اللَّهِ بْنِ عُرْوَةَ، عَنْ عُرْوَةَ، عَنْ عَائِشَةَ، قَالَتْ تَزَوَّجَنِي رَسُولُ اللَّهِ صلى الله عليه وسلم فِي شَوَّالٍ وَبَنَى بِي فِي شَوَّالٍ فَأَىُّ نِسَاءِ رَسُولِ اللَّهِ صلى الله عليه وسلم كَانَ أَحْظَى عِنْدَهُ مِنِّي ‏.‏ قَالَ وَكَانَتْ عَائِشَةُ تَسْتَحِبُّ أَنْ تُدْخِلَ نِسَاءَهَا فِي شَوَّالٍ ‏.‏

Reference	: Sahih Muslim 1423 a
In-book reference	: Book 16, Hadith 85
USC-MSA web (English) reference	: Book 8, Hadith 3312
(deprecated numbering scheme)

The Book of Fasting

(39) Chapter: It is recommended to fast Six Days in Shawwal following Ramadan

Abu Ayyub al-Ansari (Allah be pleased with him) reported Allah's Messenger (ﷺ) as saying:

He who observed the fast of Ramadan and then followed it with six (fasts) of Shawwal. it would be as if he fasted perpetually.

حَدَّثَنَا يَحْيَى بْنُ أَيُّوبَ، وَقُتَيْبَةُ بْنُ سَعِيدٍ، وَعَلِيُّ بْنُ حُجْرٍ، جَمِيعًا عَنْ إِسْمَاعِيلَ، - قَالَ ابْنُ أَيُّوبَ حَدَّثَنَا إِسْمَاعِيلُ بْنُ جَعْفَرٍ، - أَخْبَرَنِي سَعْدُ بْنُ سَعِيدِ بْنِ قَيْسٍ، عَنْ عُمَرَ، بْنِ ثَابِتِ بْنِ الْحَارِثِ الْخَزْرَجِيِّ عَنْ أَبِي أَيُّوبَ الأَنْصَارِيِّ، - رضى الله عنه - أَنَّهُ حَدَّثَهُ أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم قَالَ ‏ "‏ مَنْ صَامَ رَمَضَانَ ثُمَّ أَتْبَعَهُ سِتًّا مِنْ شَوَّالٍ كَانَ كَصِيَامِ الدَّهْرِ ‏"‏ ‏.

Reference	: Sahih Muslim 1164 a
In-book reference	: Book 13, Hadith 264
USC-MSA web (English) reference	: Book 6, Hadith 2614
(deprecated numbering scheme)

References:
http://sunnah.com/muslim/16/85
http://sunnah.com/muslim/13/264