An-Nawawi records in his book Riyad us-Saliheen:
Abdullah ibn Umar, may Allah be pleased with him, reported:
كَانَ رَسُولُ اللَّهِ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ يَعْتَكِفُ الْعَشْرَ الْأَوَاخِرَ مِنْ رَمَضَانَ
The Messenger of Allah, peace and blessings be upon him, would seek seclusion (i'tikaf) in the mosque during the last ten nights of Ramadan.
Source: Sahih Bukhari 1921, Grade: Mutaffaqun Alayhi
Aisha, may Allah be pleased with her, reported:
أَنَّ النَّبِيَّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ كَانَ يَعْتَكِفُ الْعَشْرَ الْأَوَاخِرَ مِنْ رَمَضَانَ حَتَّى تَوَفَّاهُ اللَّهُ عَزَّ وَجَلَّ ثُمَّ اعْتَكَفَ أَزْوَاجُهُ مِنْ بَعْدِهِ
The Prophet, peace and blessings be upon him, used to retreat in the mosque during the last ten nights of Ramadan until Allah the Exalted took him. His wives continued to observe this practice after him.
Source: Sahih Muslim 1172, Grade: Sahih
Abu Huraira, may Allah be pleased with him, reported:
كَانَ النَّبِيُّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ يَعْتَكِفُ فِي كُلِّ رَمَضَانٍ عَشْرَةَ أَيَّامٍ فَلَمَّا كَانَ الْعَامُ الَّذِي قُبِضَ فِيهِ اعْتَكَفَ عِشْرِينَ يَوْمًا
The Prophet, peace and blessings be upon him, would seek seclusion in the mosque for ten days during every Ramadan, and in the year he passed away he sought seclusion for twenty days.
Source: Sahih Bukhari 1939, Grade: Sahih
Imam Ash-Shafi'ee, may Allah have mercy on them, said:
وَمَنْ أَرَادَ الِاقْتِدَاءَ بِالنَّبِيِّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ فِي الِاعْتِكَافِ فِي الْعَشْرِ الْأَوَاخِرِ مِنْ رَمَضَانَ فَيَنْبَغِي أَنْ يَدْخُلَ الْمَسْجِدَ قَبْلَ غُرُوبِ الشَّمْسِ لَيْلَةَ الْحَادِي وَالْعِشْرِينَ مِنْهُ
Whoever desires to follow the example of the Prophet, peace and blessings be upon him, should seek seclusion in the mosque during the last ten nights of Ramadan; so let him enter the mosque before the setting of the sun on the twenty first night.
Source: Al-Majmu An-Nawawi 501