Shaking hands was a custom that was well known among the Sahaabah (may Allaah be pleased with them).
It was narrated that Qataadah said: I said to Anas ibn Maalik: Did the companions of the Messenger of Allaah (peace and blessings of Allaah be upon him) shake hands with one another? He said: Yes. Narrated by al-Bukhaari (6263).
Ibn Battaal said: Shaking hands is good according to most of the scholars. Al-Nawawi said: Shaking hands when meeting is a Sunnah on which there is consensus – as stated in Fath al-Baari (11/55).
Shaking hands happens when one man places his hand in the hand of his companion; this is what is implied by the Arabic word musaafahah (handshaking), as stated in Mu’jam Maqaayees al-Lughah (3/229) and elsewhere. Based on this the apparent meaning of the ahaadeeth quoted above about handshaking may be understood.
Hence most of the scholars are of the view that shaking hands with one hand is the Sunnah, and this is the usual custom among the Muslims and the Sahaabah (may Allaah be pleased with them).
Shaykh al-Albaani said in al-Silsilah al-Saheehah (1/22), mentioning what is to be learned from some ahaadeeth:
Taking hold of one hand when shaking hands. This is mentioned in many ahaadeeth, and it is what is implied linguistically.
I say: In some of the ahaadeeth referred to this meaning may also be understood, such as the marfoo’ hadeeth of Hudhayfah: “When the believer meets a fellow believer and he greets him with salaam and takes him by the hand and shakes hands with him, their sins will fall like the leaves of a tree.” Al-Mundhiri said (3/270): It was narrated by al-Tabaraani in al-Awsat and I do not know of anyone who is majrooh (unacceptable) among its narrators. I say: And it has corroborating evidence which raises it to the level of being saheeh.
These ahaadeeth all indicate that the Sunnah in shaking hands is to use just one hand. End quote.
As for the view of some Hanafi and Maaliki fuqaha’, that it is mustahabb to shake hands using both hands, putting the palm of the left hand over the back of one’s brother’s hand, this is not proven to be Sunnah from the Prophet (peace and blessings of Allaah be upon him) or from his companions. Rather the most that can be said concerning some ahaadeeth that refer to it is that the Prophet (peace and blessings of Allaah be upon him) took the hand of one of his companions in both of his hands as a sign of extra care in teaching, guiding and so on, as it says in Saheeh al-Bukhaari (6265) and Saheeh Muslim (402) that Ibn Mas’ood (may Allaah be pleased with him) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) taught me the tashahhud, holding my hand between both of his.
But that is not the usual habit, based on the evidence of the previous report which says that the basic principle is shaking hands with one hand, which is stated clearly in some reports. This hadeeth indicates that too, because if the custom among the Sahaabah had been to shake hands using both hands, Ibn Mas’ood would not have mentioned that. The fact that he mentions it indicates that it was not the usual habit of the Prophet (peace and blessings of Allaah be upon him) with his companions.
Nevertheless, shaking hands using both hands should not be described as a bid’ah (innovation), rather it is something that is permissible, but the Sunnah, which it is better to adhere to, is to shake hands using just one hand.
It was narrated from Hammaad ibn Zayd that he shook hands with ‘Abd-Allaah ibn al-Mubaarak using both hands, as is stated in a mu’allaq report in Saheeh al-Bukhaari (p. 1206).
It says in Fataawa al-Lajnah al-Daa’imah (24/125):
As for shaking hands using both hands, we do not know anything about that, but it should not be done; it is better to use just one hand.
See also al-Mawsoo’ah al-Fiqhiyyah: entry on Musaafahah (shaking hands); Tuhfat al-Ahwadhi (7/431-433)
And Allaah knows best.