All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is a recommended Sunnah that men shake hands with men and their female Mahram, and women shake hands with women and thier male Mahram (unmarriageable relative). Ahmad, Abu Daawood, An-Nasaa'i, and Ibn Maajah reported that the Prophet said: "No two Muslims meet and shake hands with each other, except that Allaah forgives them before they leave each other." Ahmad and At-Tirmithi reported that a man came to the Prophet and said: "One of us meets the other, does he bow to him. The Prophet said: "No." The man said: 'Does he hug him and kiss him.' He said: "No." Then the man asked: 'Does he take his hand and shake hands with him?' The Prophet said: "Yes."
However, the fact that the Prophet told him not to hug each other is limited to people who are not coming from traveling. The evidence is the narration of Anas who said: "The companions of the Prophet used to shake hands whenever they meet each other and they used to embrace each other when they came from a journey." [At-Tabaraani]
Imaam An-Nawawi said: "Shaking hands at every meeting is a Sunnah." Ibn Rushd said: "It is recommended."
However, according to the Sunnah, shaking hands should be with the right hand only, not with both hands for the following reasons:
1. This was the usual practice during the period of the Prophet . Imaam Muslim reported that 'Amr Ibn Al-'Aas said: "I came to the Prophet and said: "Stretch out your right hand …." So, the Prophet stretched his right hand only but he did not stretch both hands. This is also what 'Amr Ibn Al-'Aas asked him to do.
Moreover, Imaam Ahmad reported that Abu Ghaadiyah said: "I gave the pledge of allegiance to the Messenger of Allaah. Abu Sa'eed asked him, 'Did you really shake his hand with your right hand?' He said "Yes." Imaam Ahmad also reported that Anas said: "I gave my pledge of obedience to the Prophet with my hand, i.e. his right hand."
2. Linguistically Musaafahah (shaking hands) in Arabic means touching the palms of the hand, it does not mean touching the back of the hand of the person with the left hand while shaking hands with the right hand.
3. The left hand is not usually used in good and respected actions, rather it is used to remove impurity, or to sniff water, and the like. Therefore, it is not suitable to use it in shaking hands even if it is along with the right hand. 'Aa'ishah said: "The Prophet gave preference to his right side in all his matters; for putting on his shoes, for combing his hair and for performing ablution".
Mubaarak Poori said: "Shaking hands or giving a pledge of allegiance with both hands is not confirmed in any sound, clear and authentic narration."
However, if one shakes hands with both hands he does not commit any sin. Besides, there is no specific supplication that is said at the time of shaking hands. Some scholars are of the opinion that one should ask forgiveness of Allaah at that time, but this is not confirmed to be from the Sunnah of the Prophet .
Allaah Knows best.