Is it forbidden for Muslims men and women to shake hands ?

Shaking hands (and touching) members of the opposite sex when not related, is not permissible for Muslims according to the teachings of Islam.

  It is not allowed for a believing man to put his hand in the hand of a woman who is not allowed for him (mother, wife, sister, daughter, etc.). Whoever does this has wronged himself (sinned).

There is a Hadith from Ma'qil ibn Yasar, saying; The Prophet (Peace and blessings be upon him) said, "It is better for you to be stabbed in the head with an iron needle than to touch the hand of a woman who is not permissible to you."

This alone should be enough to keep away from this action and to instill obedience to Allah, as it implies touching women may lead to temptation and immorality.

Ayesha, the wife of the Prophet (Peace and blessings be upon him) said: 'When the believing women migrated (to Medina) and came to the Prophet (Peace and blessings be upon him) they would be examined in accordance with the words of Allah (in Quran):

"O Prophet! When believing women come to you to give you the Bay’ah (pledge of allegiance), that they will not associate anything in worship with Allah, that they will not steal, that they will not commit illegal sexual intercourse, that they will not kill their children, that they will not utter slander, intentionally forging falsehood (i.e. by making illegal children belonging to their husbands), and that they will not disobey you in any Ma'ruf (Islamic Monotheism and all that which Islam ordains) then accept their Bay’ah (pledge of allegiance), and ask Allah to forgive them, Verily, Allah is Oft-Forgiving, Most Merciful". [Noble Quran 60:12]

Any of the believing women who accepted the conditions of the verse and agreed to live by them were considered to have offered themselves for giving their oaths of allegiance. When they declared their commitment to do so, the Prophet (Peace and blessings be upon him) would say to them, "You may go. I have confirmed your allegiance."
I swear by Allah, the Prophet's hand never touched the hand of a woman. He would receive their oath of allegiance by spoken declaration. I swear by Allah, the Prophet (Peace and blessings be upon him) never took any vow from women except what Allah had ordered him to take and his palm never touched the palm of a woman. When he had taken their pledge, he would tell them he had taken their oath from them orally. ' [Sahih Muslim]

The Prophet of Allah (Peace and blessings be upon him) did not touch women who were not permissible (shaking hands, etc.). This despite the fact the oath of allegiance was originally given by hand. So what about these other men going around shaking hands?

Umaymah bint Raqiqah said: 'The Prophet (Peace and blessings be upon him) said, "I do not shake hands with women (not permissible to touch)."'

May Allah guide us to the straight path...
Aameen ya Rabb...