Answer
Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
There is nothing wrong with shaking hands with a mother-in-law because she is a mahram,
Answering your question, Sheikh Salih Ibn Fawzan Al-Fawzan, A member of the Permanent Committee for Scholarly Research and Ifta’ in Saudi Arabia, states:
Yes, there is nothing wrong with shaking hands with wife’s mother because she is a mahram, as Allah has made her permanently unlawful for marriage to the husband of her daughter.
Allah says, listing the women who are mahrams: “your wives’ mothers” (An-Nisaa’ 4:23) i.e., your wives’ mothers are mahrams (forbidden in marriage) to you.
Therefore she is one of your mahrams, and there is nothing wrong with you shaking hands with her or traveling with her.
But if there is the fear of fitnah, then you should not shake hands with her, such as if you fear that there will be some temptation or provocation of desire in shaking hands with her, then do not shake hands with her. (Al-Muntaqa min Fatawa al-Shaykh al-Fawzan 3/question no.450)
Allah Almighty knows best.
Source: www.islamqa.info/en