In the name of Allah, the Most Merciful, the Bestower of Mercy
Aboo ‘Uthmaan an-Nahdi said:
The Salaf (pious predecessors) used to venerate three tens of (days): The last ten days of Ramadaan, the first ten days of Dhul-Hijjah and the first ten days of Muharram. 
Al-Haafidh ibn Hajr al-‘Asqalaanee said;
It appears that the preference of these [ten] days is due to the gathering of the main acts of worship in Dhul-Hijjah are praying, fasting, giving charity, and pilgrimage which are restricted to these days. 
Shaykh-ul-Islaam was asked the following question: “which is better, the first ten days of Dhul-Hijjah or the last ten days of Ramadan?” He answered saying:
The first ten days of Dhul-Hijjah are better than the last ten days of Ramadan, but the last ten nights of Ramadan is better than the first ten nights of Dhul-Hijjah. 
Al-Haafidh Ibn Rajab said,
What is correct is that the first ten days and nights of Dhul-Hijjah is better than the last ten days of Ramadan except for the night of laylat al-Qadr. 
From those who used to fast the ten days was Abdullaah ibn Umar -radiyaAllaahu anhumaa- and it was stated most of the ulamaah or majority of them recommend the fasting of these days.  
“Fasting these days is highly recommended. There are no days like it, and that is the day of Arafah.” 
Al-Haafidh ibn Rajab said:
This Hadeeth has shown that deeds in the (first ten) days of Dhul-Hijjah are more beloved to Allaah than deeds in any other days of this Dunya without any exception to that. If they are more beloved to Allaah then they are more virtuous with Him. This Hadeeth has been narrated with the wording, ‘There are no days in which deeds are better than the ten days’ It was narrated with some doubts in the wording as to whether it was ‘more beloved’ or ‘better’. If the deeds in the ten days [of Dhul-Hijjah] are better and more beloved to Allaah than deeds in the rest of the entire year, then deeds performed within those days – even if they are superior.
Concerning this they (the Companions) said,
O Messenger of Allaah, Not even Jihaad in the path of Allaah?
Not even Jihaad in the path of Allaah
He then made the exception of one type of Jihaad which is the best of Jihaad…And as for the remaining types of Jihaad, then definitely deeds in the [first] ten of Dhul-Hijjah are better and more beloved to Allaah – Azza wa Jal than that.” 
Al-Haafidh Ibn Rajab said,
It is recommended to stand the nights in prayer (Qiyaam Al-layl) during the first ten days of Dhul-Hijjah. Imaam ash-Shaafi’ee, as well as other scholars, have said that it is recommended to do so.
Sa’eed ibn Jubayr narrated a hadith in which he said that whenever the first ten days of Dhul-Hijjah began, Ibn ‘Abbas (radiyaAllaahu anhu) would exert himself to the limits of his ability.
It has also been narrated upon (Ibn Abbaas) that he said,
Do not turn out your lamps during the nights of the first ten days of Dhul-Hijjah
This is because he was especially fond of worship (during those days).” 
 Narrated by Anas Ibn Malik; Collected by Tirmidhee & Ibn Maajah
 Fath-ul-Baari 2/534
 Majmoo’ fatawaa ibn taymiyyah (25/154)
 Lataa’if al-Maarif pg.522
 Translater Note: Fasting is only up to the 9th of Dhul-Hijjah. The 10th is on Eid-Adha which is not permissible to fast.
 Sharh Saheeh Muslim 4/328
 Lataa’if al-Ma’aarif (520-521)
 Lataa’if-ul-Ma’aarif (462)