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

الترغيب في المصافحة

Encouragement of shaking hands.

Source: al-Targhīb wal-Tarhīb 3/431

Al-Bara’ ibn ‘Azib reported: The Messenger of Allah, peace and blessings be upon him, said:

مَا مِنْ مُسْلِمَيْنِ يَلْتَقِيَانِ فَيَتَصَافَحَانِ إِلَّا غُفِرَ لَهُمَا قَبْلَ أَنْ يَفْتَرِقَا

No two Muslims meet each other and shake hands but that Allah forgives them before they part.

Source: Sunan al-Tirmidhī 2727, Grade: Sahih

Anas ibn Malik, may Allah be pleased with him, said:

كَانَ أَصْحَابُ النَّبِيِّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ إِذَا تَلاقَوْا تَصَافَحُوا وَإِذَا قَدِمُوا مِنْ سَفَرٍ تَعَانَقُوا

If the companions of the Prophet, peace and blessings be upon him, met each other, they would shake hands. If they had returned from a journey, they would embrace each other.

Source: al-Mu’jam al-Awsaṭ 97, Grade: Jayyid

Hudhayfah ibn al-Yaman reported: The Prophet, peace and blessings be upon him, said:

إِنَّ الْمُؤْمِنَ إِذَا لَقِيَ الْمُؤْمِنَ فَسَلَّمَ عَلَيْهِ وَأَخَذَ بِيَدِهِ فَصَافَحَهُ تَنَاثَرَتْ خَطَايَاهُمَا كَمَا يَتَنَاثَرُ وَرَقُ الشَّجَرِ

Verily, when the believer meets another believer, greets him with peace, and shakes his hand, the sins of them both will shed like leaves falling from a tree.

Source: al-Muʻjam al-Awsaṭ 253, Grade: Sahih

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

إِنَّ الْمُسْلِمَ إِذَا صَافَحَ أَخَاهُ تَحَاتَّتْ خَطَايَاهُمَا كَمَا يَتَحَاتُّ وَرَقُ الشَّجَرِ

Verily, when the Muslim shakes hands with his brother, both of their sins fall away like the falling of leaves from a tree.

Source: Musnad al-Bazzār 8335, Grade: Sahih li ghayrihi

Qatadah reported: I said to Anas ibn Malik, “Did the companions of the Messenger of Allah, peace and blessings be upon him, shake hands?” Anas said:

نَعَمْ

Yes.

Source: Sunan al-Tirmidhī 2729, Grade: Sahih