Al-ʿAllāmah Muḥammad Nāṣir al-Dīn al-Albānī said:
From among the widespread innovations of this time—of which none has the capability to point out except the one who is well acquainted with the sunnah of the Prophet (ﷺ) and the Salaf al-Ṣālih—is that when a person enters and gives his salām, shakes hands (and from the perfection of the greetings is to shake hands even though the narration it comes in is weak but its meaning is correct), so he shakes hands but what is incorrect is that he gives salām [individually] to everyone he shakes hands with. So he says: “salāmu ʿalaykum, salāmu ʿalaykum, salāmu ʿalaykum” if there are twenty people in the gathering he gives twenty salāms; this is an innovation.
Rather, the Sunnah is that when a person enters a gathering he say, “salāmu ʿalaykum” once, after this if it is possible for him to shake hands with those present in the gathering then that is better.
If it is not possible for him to shake hands, he has accomplished the obligation of the Muslim giving salām to another Muslim—one-fifth (of the rights a Muslim has upon a Muslim)—that if you come upon another Muslim, you give him salām. As for shaking hands, this is a recommended Sunnāh.
As some of the companions have said:
We have not come upon the Messenger of Allāh (ﷺ) except that we shook his hand.
So adding to the Sunnah al-Qawliyyah (the Prophet’s speech) or Sunnah al-Fiʿliyyah (the Prophet’s actions) is an innovation in its essence. For this reason, if it is said that adding good [to the sunnah] is good, [it is said] this speech will not save you because if it was truly good we would have been preceded in it (by the early generations). This is my advice.
Source: Silsilat al-Hudā wa-al-Nūr #181
Translated by: Munīb al-Ṣumālī