Al-Mundhiri recorded the following chapter in his book Al-Targhib wal-Tarhib:

الترغيب في الحياء وما جاء في فضله والترهيب من الفحش والبذاء

Inspiring modesty and what has been related of its virtues, and discouraging obscenity and shamelessness.

Source: al-Targhīb wal-Tarhīb 3/397

Ibn Umar reported: The Prophet, peace and blessings be upon him, passed by a man from the Ansar who was admonishing his brother about modesty. The Prophet said:

دَعْهُ فَإِنَّ الْحَيَاءَ مِنَ الإِيمَانِ

Let him be, for modesty is part of faith.

Source: Ṣaḥīḥ al-Bukhārī 24, Grade: Muttafaqun Alayhi

‘Imran ibn Husayn reported: The Prophet, peace and blessings be upon him, said:

الْحَيَاءُ لَا يَأْتِي إِلَّا بِخَيْرٍ

Modesty does not bring anything but goodness.

Source: Ṣaḥīḥ al-Bukhārī 5766, Grade: Muttafaqun Alayhi

In another narration, the Prophet said:

الْحَيَاءُ خَيْرٌ كُلُّهُ

Modesty is all that is good.

Source: Ṣaḥīḥ Muslim 37, Grade: Sahih

Abu Huraira reported: The Messenger of Allah, peace and blessings be upon him, said:

الْإِيمَانُ بِضْعٌ وَسَبْعُونَ أَوْ بِضْعٌ وَسِتُّونَ شُعْبَةً فَأَفْضَلُهَا قَوْلُ لَا إِلَهَ إِلَّا اللَّهُ وَأَدْنَاهَا إِمَاطَةُ الْأَذَى عَنْ الطَّرِيقِ وَالْحَيَاءُ شُعْبَةٌ مِنْ الْإِيمَانِ

Faith has sixty to seventy branches, the best of which is to declare there is no God but Allah, the least of which is to remove something harmful from the road, and modesty is a branch of faith.

Source: Ṣaḥīḥ al-Bukhārī 9, Grade: Muttafaqun Alayhi

Abu Huraira reported: The Messenger of Allah, peace and blessings be upon him, said:

الْحَيَاءُ مِنْ الْإِيمَانِ وَالْإِيمَانُ فِي الْجَنَّةِ وَالْبَذَاءُ مِنْ الْجَفَاءِ وَالْجَفَاءُ فِي النَّارِ

Modesty is part of faith and faith is in Paradise. Shamelessness is part of insolence and insolence is in Hellfire.

Source: Sunan al-Tirmidhī 2009, Grade: Sahih

Abu Umamah reported: The Prophet, peace and blessings be upon him, said:

الْحَيَاءُ وَالْعِيُّ شُعْبَتَانِ مِنْ الْإِيمَانِ وَالْبَذَاءُ وَالْبَيَانُ شُعْبَتَانِ مِنْ النِّفَاقِ

Modesty and reticence are two branches of faith. Profanity and temerity are two branches of hypocrisy.

Source: Sunan al-Tirmidhī 2027, Grade: Sahih

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Ibn Umar reported: The Prophet, peace and blessings be upon him, said:

إِنَّ الْحَيَاءَ وَالإِيمَانَ قُرِنَا جَمِيعًا فَإِذَا رُفِعَ أَحَدُهُمَا رُفِعَ الآخَرُ

Verily, modesty and faith come together. If one of the two is missing, so is the other.

Source: Shu’ab al-Imān 7211, Grade: Sahih

Abdullah ibn Mas’ud reported: The Messenger of Allah, peace and blessings be upon him, said:

اسْتَحْيُوا مِنْ اللَّهِ حَقَّ الْحَيَاءِ

Be modest before Allah, as is his right.

We said, “O Messenger of Allah, surely we have modesty, all praise is due to Allah!” The Prophet said:

لَيْسَ ذَاكَ وَلَكِنَّ الِاسْتِحْيَاءَ مِنْ اللَّهِ حَقَّ الْحَيَاءِ أَنْ تَحْفَظَ الرَّأْسَ وَمَا وَعَى وَالْبَطْنَ وَمَا حَوَى وَلْتَذْكُرْ الْمَوْتَ وَالْبِلَى وَمَنْ أَرَادَ الْآخِرَةَ تَرَكَ زِينَةَ الدُّنْيَا فَمَنْ فَعَلَ ذَلِكَ فَقَدْ اسْتَحْيَا مِنْ اللَّهِ حَقَّ الْحَيَاءِ

It is not so. Rather, modesty before Allah, as is his right, is to guard the mind and what runs through it, to guard the stomach and what fills it, and to reflect upon death and trials. Whoever desires the Hereafter, let him abandon the embellishment of worldly life. Whoever does so has been modest before Allah, as is his right.

Source: Sunan al-Tirmidhī 2458, Grade: Sahih