Evidences
Anas bin Malik said, "Any of the female slaves of Medina could take hold of the hand of Allah's Messenger (saw) and take him wherever she wished." (Bukhari)
Anas bin Malik said: “If a female slave among the people of Medinah were to take the hand of the Messenger of Allah (saw), he would not take his hand away from hers until she had taken him wherever she wanted in Medinah so that her needs may be met.” (Ibn Majah, Ahmed)
The Messenger of Allah (saw) used to visit Umm Haram, daughter of Milhan. She was the wife of Ubada bin Samit, One day the Messenger of Allah (saw) paid her a visit. She entertained him with food and then sat down to rub his head. (Bukhari, Muslim)
Abu Musa al-Ashari (ra) whilst on Hajj went to a woman of al-Qais who removed lice from his head (Bukhari)
Ibrahim al-Nikhae said, "I met a woman and I intended to shake my hands with her. I placed my hand over the cloth and she took of her niqab, She was one of the woman of my area who became middle aged. I shook my hand with her and I didn't have anything on my hand." (حلية الأولياء 2/442)
Aisha (ra) said: "The Prophet's hands never touched those of a woman" (Muslim and Bukhari).
The Messenger of Allah (saw) said: “For one of you to be stabbed in the head with an iron needle is better for him than that he should touch a woman who is not permissible for him.” (Tabarani)
The Messenger of Allah (saw) said: "If anyone of you to be stabbed by iron needle on its head its dearer to me than your head be washed off by a woman who is not mahram to you." (Bukhari, Muslim)
The Messenger of Allaah (saw) said, “I do not shake hands with women.” (Nisa’i)
Analysis
Those prohibiting shaking of hands argue instances where hadiths narrate the messenger shook hands with women are specific to him. Qadhi Iyaz refuted such claims by arguing khasais (those things which are specific to only prophet) cannot be proven unless there is a clear proof for it, as the general principle established from texts is all his actions can be emulated by his ummah.
Furthermore narrations that companions engaged in such acts counter such claims.
Narrations that appear to prohibit touching a non-mahram use the term mass (touch) to metaphorically refer to sexual acts so do Not apply to the issue of shaking hands.
Conclusion
Shaking hands between men and women who are not mahram without desire is permitted by some jurists - most however prohibit the practice.
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