Fasting six days of Shawwal after the obligatory fast of Ramadan is Sunnah Mustahabbah, not wajib. It is recommended for the Muslim to fast six days of Shawwal, and in this there is great virtue and an immense reward.
Abu Ayub narrated that, the Messenger of Allah said: "Whoever fasts Ramadan, then follows it with six from Shawwal, then that is (equal in reward) to fasting everyday." (Grade-Sahih)
Reference: Jami' at-Tirmidhi 759; Book 8, Hadith 78; Vol. 2, Book 3, Hadith 759
Whoever fasts these six days will have recorded for him a reward as if he had fasted a whole year, as was reported in a sahih hadith from the Prophet (peace and blessings of Allah be upon him). It was narrated from Thawban, the freed slave of the Messenger of Allah (ﷺ), that the Messenger of Allah (ﷺ) said: “Whoever fasts six days after the Fitr will have completed the year, for whoever does a good deed will have the reward of ten like it.” (Grade: Sahih)
حَدَّثَنَا هِشَامُ بْنُ عَمَّارٍ، حَدَّثَنَا بَقِيَّةُ، حَدَّثَنَا صَدَقَةُ بْنُ خَالِدٍ، حَدَّثَنَا يَحْيَى بْنُ الْحَارِثِ الذِّمَارِيُّ، قَالَ سَمِعْتُ أَبَا أَسْمَاءَ الرَّحَبِيَّ، عَنْ ثَوْبَانَ، مَوْلَى رَسُولِ اللَّهِ ـ صلى الله عليه وسلم ـ عَنْ رَسُولِ اللَّهِ ـ صلى الله عليه وسلم ـ أَنَّهُ قَالَ " مَنْ صَامَ سِتَّةَ أَيَّامٍ بَعْدَ الْفِطْرِ كَانَ تَمَامَ السَّنَةِ {مَنْ جَاءَ بِالْحَسَنَةِ فَلَهُ عَشْرُ أَمْثَالِهَا} " .
Reference : Vol. 1, Book 7, Hadith 1715; Book 7, Hadith 1786
According to another report, Abu Dharr (May Allah be pleased with him) reported: The Prophet (ﷺ) said, "Allah, the Almighty, says: 'Whosoever does a good deed, will have (reward) ten times like it and I add more; and whosoever does an evil, will have the punishment like it or I will forgive (him); and whosoever approaches Me by one span, I will approach him by one cubit; and whosoever approaches Me by one cubit, I approach him by one fathom, and whosoever comes to Me walking, I go to him running; and whosoever meets Me with an earth-load of sins without associating anything with Me, I meet him with forgiveness like that". [Muslim]
وعن أبي ذر، رضي الله عنه ، قال: قال النبي صلى الله عليه وسلم : "يقول الله عز وجل: من جاء بالحسنة، فله عشر أمثالها أو أزيد، ومن جاء بالسيئة، فجزاء سيئة سيئة مثلها أوغفر. ومن تقرب مني شبراً ، تقربت منها ذراعا، ومن تقرب منى ذراعاً ، تقربت منه باعاً، ومن أتاني يمشي ، أتيته هرولة، ومن لقينى بقراب الأرض خطيئة لايشرك به شيئاً ، لقيته بمثلها مغفرة" ((رواه مسلم)).
Reference : Riyad-us-salihin Book 1, Hadith 413
Another of the important benefits of fasting six days of Shawwal is that it makes up for any shortfall in a person's obligatory Ramadan fasts, because no one is free of shortcomings or sins that have a negative effect on his fasting. On the Day of Resurrection, some of his nafil deeds will be taken to make up the shortcomings in his obligatory deeds, as It was narrated from Abu Hurairah that the Prophet (ﷺ) said: "The first thing for which a person will be brought to account on the Day of Resurrection will be his Salah. If it is found to be complete then it will be recorded as complete, and if anything is lacking He will say: 'Look and see if you can find any voluntary prayers with which to complete what he neglected of his obligatory prayers.' Then the rest of his deeds will be reckoned in like manner." (Grade: Sahih)
أَخْبَرَنَا أَبُو دَاوُدَ، قَالَ حَدَّثَنَا شُعَيْبٌ، - يَعْنِي ابْنَ زِيَادِ بْنِ مَيْمُونٍ - قَالَ كَتَبَ عَلِيُّ بْنُ الْمَدِينِيِّ عَنْهُ - أَخْبَرَنَا أَبُو الْعَوَّامِ، عَنْ قَتَادَةَ، عَنِ الْحَسَنِ، عَنْ أَبِي رَافِعٍ، عَنْ أَبِي هُرَيْرَةَ، أَنَّ النَّبِيَّ صلى الله عليه وسلم قَالَ " إِنَّ أَوَّلَ مَا يُحَاسَبُ بِهِ الْعَبْدُ يَوْمَ الْقِيَامَةِ صَلاَتُهُ فَإِنْ وُجِدَتْ تَامَّةً كُتِبَتْ تَامَّةً وَإِنْ كَانَ انْتَقَصَ مِنْهَا شَىْءٌ قَالَ انْظُرُوا هَلْ تَجِدُونَ لَهُ مِنْ تَطَوُّعٍ يُكَمِّلُ لَهُ مَا ضَيَّعَ مِنْ فَرِيضَةٍ مِنْ تَطَوُّعِهِ ثُمَّ سَائِرُ الأَعْمَالِ تَجْرِي عَلَى حَسَبِ ذَلِكَ "
Reference: Sunan an-Nasa'i 466; Book 5, Hadith 19; Vol. 1, Book 5, Hadith 467
It was also narrated from Abu Hurayrah (may Allah be pleased with him) who said: The Messenger of Allah (peace and blessings of Allah be upon him) said, “Verily Allah ta’ala has said: ‘Whosoever shows enmity to a wali (friend) of Mine, then I have declared war against him. And My servant does not draw near to Me with anything more loved to Me than the religious duties I have obligated upon him. And My servant continues to draw near to me with nafil (supererogatory) deeds until I Love him. When I Love him, I am his hearing with which he hears, and his sight with which he sees, and his hand with which he strikes, and his foot with which he walks. Were he to ask [something] of Me, I would surely give it to him; and were he to seek refuge with Me, I would surely grant him refuge.’ ” [Al-Bukhari]
عَنْ أَبِي هُرَيْرَة رَضِيَ اللهُ عَنْهُ قَالَ: قَالَ رَسُول اللَّهِ صلى الله عليه و سلم إنَّ اللَّهَ تَعَالَى قَالَ: "مَنْ عَادَى لِي وَلِيًّا فَقْد آذَنْتهُ بِالْحَرْبِ، وَمَا تَقَرَّبَ إلَيَّ عَبْدِي بِشَيْءٍ أَحَبَّ إلَيَّ مِمَّا افْتَرَضْتُهُ عَلَيْهِ، وَلَا يَزَالُ عَبْدِي يَتَقَرَّبُ إلَيَّ بِالنَّوَافِلِ حَتَّى أُحِبَّهُ، فَإِذَا أَحْبَبْتُهُ كُنْت سَمْعَهُ الَّذِي يَسْمَعُ بِهِ، وَبَصَرَهُ الَّذِي يُبْصِرُ بِهِ، وَيَدَهُ الَّتِي يَبْطِشُ بِهَا، وَرِجْلَهُ الَّتِي يَمْشِي بِهَا، وَلَئِنْ سَأَلَنِي لَأُعْطِيَنَّهُ، وَلَئِنْ اسْتَعَاذَنِي لَأُعِيذَنَّهُ". [رَوَاهُ الْبُخَارِيُّ].
Reference: 40 Hadith Nawawi 38; Hadith 38
Moreover, It is not a necessary condition that they should be fasted consecutively. If you fast them separately or consecutively, it is OK. The sooner you do them, the better, because Allah says (interpretation of the meanings):
وَاَنۡزَلۡنَاۤ اِلَيۡكَ الۡكِتٰبَ بِالۡحَـقِّ مُصَدِّقًا لِّمَا بَيۡنَ يَدَيۡهِ مِنَ الۡكِتٰبِ وَمُهَيۡمِنًا عَلَيۡهِ فَاحۡكُمۡ بَيۡنَهُمۡ بِمَاۤ اَنۡزَلَ اللّٰهُ وَلَا تَتَّبِعۡ اَهۡوَآءَهُمۡ عَمَّا جَآءَكَ مِنَ الۡحَـقِّؕ لِكُلٍّ جَعَلۡنَا مِنۡكُمۡ شِرۡعَةً وَّمِنۡهَاجًا ؕ وَلَوۡ شَآءَ اللّٰهُ لَجَـعَلَـكُمۡ اُمَّةً وَّاحِدَةً وَّلٰـكِنۡ لِّيَبۡلُوَكُمۡ فِىۡ مَاۤ اٰتٰٮكُمۡ فَاسۡتَبِقُوا الۡخَـيۡـرٰتِؕ اِلَى اللّٰهِ مَرۡجِعُكُمۡ جَمِيۡعًا فَيُنَبِّئُكُمۡ بِمَا كُنۡتُمۡ فِيۡهِ تَخۡتَلِفُوۡنَۙ
And We have revealed to you, [O Muhammad], the Book in truth, confirming that which preceded it of the Scripture and as a criterion over it. So judge between them by what Allah has revealed and do not follow their inclinations away from what has come to you of the truth. To each of you We prescribed a law and a method. Had Allah willed, He would have made you one nation [united in religion], but [He intended] to test you in what He has given you; so race to [all that is] good. To Allah is your return all together, and He will [then] inform you concerning that over which you used to differ. (Surah 5:48)
وَسَارِعُوۡۤا اِلٰى مَغۡفِرَةٍ مِّنۡ رَّبِّكُمۡ وَجَنَّةٍ عَرۡضُهَا السَّمٰوٰتُ وَالۡاَرۡضُۙ اُعِدَّتۡ لِلۡمُتَّقِيۡنَۙ
And hasten to the forgiveness of your Lord and to a Paradise as vast as the heavens and the earth, prepared for the God-fearing. (Surah 3:133)
And (you should hasten to fast these six days) because delaying may cause problems. This is the view of the Shafi'is and some of the Hanbalis, but it is OK if you do not hasten it and you delay it until the middle or end of the month.
An-Nawawi (may Allah have mercy on him) said:
"Our companions said: it is mustahabb to fast six days of Shawwal. Because of this hadith they said: it is mustahabb to fast these days consecutively at the beginning of Shawwal, but if one separates them or delays them until after Shawwal, this is permissible, because he will still be following the general guidelines of the hadith. We have no dispute regarding this matter, and this was also the view of Ahmad and Dawud."Al-Majmu' Sharh al-Muhadhdhab
Furthermore "... although the majority of the scholars didn't obligate the six days of Shawwal to have to be immediate, or six consecutive days, it is easier and safer to do it that way and you will avoid any differences between the scholars that way. However, the matter is flexible, so whenever you can."
The precise rewards for the deeds which people do for the sake of Allah is something which is known only to Allah. If a person seeks the reward from Allah and strives to obey Him, his reward will not be lost, as Allah says (interpretation of the meaning):
"We shall not make the reward of anyone who does his (righteous) deeds in the most perfect manner to be lost."Al-Qur'an 18:30
If someone has missed some of the days of Ramadan, he should fast them first, then fast six days of Shawwal, because he cannot follow the fast of Ramadan with six days of Shawwal unless he has completed his Ramadhan fast.
Furthermore:
if there are too many and it will push you over and you will miss Shawwal, don't worry. As soon as you are finished then fast the "six days of Shawwal" whatever the date is and you will get the reward for it, insha'Allah, because of the necessity. This was the fatwa of Shaykh Ibn Al-'Uthaymin (rahimahullah); and
if you think you will find it difficult to make up all your obligatory fasts now, especially because of the length of day and heat, then leave them for later in the year and fast the six days of Shawwal now. This was performed by 'A'ishah (radhiallahu 'anha) and despite the discussion over her action, her example can be used as an evidence for anyone who cannot fast the obligatory qadha' now, and will put the voluntary fasts before the obligatory.
وَاِذۡ يَرۡفَعُ اِبۡرٰهٖمُ الۡقَوَاعِدَ مِنَ الۡبَيۡتِ وَاِسۡمٰعِيۡلُؕ رَبَّنَا تَقَبَّلۡ مِنَّا ؕ اِنَّكَ اَنۡتَ السَّمِيۡعُ الۡعَلِيۡمُ - رَبَّنَا وَاجۡعَلۡنَا مُسۡلِمَيۡنِ لَـكَ وَ مِنۡ ذُرِّيَّتِنَآ اُمَّةً مُّسۡلِمَةً لَّكَ وَاَرِنَا مَنَاسِكَنَا وَتُبۡ عَلَيۡنَا ۚ اِنَّكَ اَنۡتَ التَّوَّابُ الرَّحِيۡمُ - رَبَّنَا وَابۡعَثۡ فِيۡهِمۡ رَسُوۡلًا مِّنۡهُمۡ يَتۡلُوۡا عَلَيۡهِمۡ اٰيٰتِكَ وَيُعَلِّمُهُمُ الۡكِتٰبَ وَالۡحِكۡمَةَ وَ يُزَكِّيۡهِمۡؕ اِنَّكَ اَنۡتَ الۡعَزِيۡزُ الۡحَكِيۡمُ -
Recall when Abraham and Ishmael raised the foundations of the House, praying: “Our Lord! Accept this from us; You are All-Hearing, All-Knowing. Our Lord! Make us submissive to You and make out of our descendants a community that submits itself to You, and show us the ways of Your worship, and turn to us in mercy. You are Much-Relenting, Most Compassionate. Our Lord! Raise up in the midst of our offspring a Messenger from among them who shall recite to them Your verses, and instruct them in the Book and in Wisdom, and purify their lives.128 Verily, You are the Most Mighty, the Most Wise.”
Rest Allah knows best.
Source: Quran, Sunnah, Sunnahonline, Islamicstudies