As far as I can remember, it is haram for men to shake hands with women who are Muslim. But I'd like to know if it is haram for Muslim men to shake hands with non-Muslim women as well?
بسم الله الرحمن الرحیم
In regard to your question that asked
What is the ruling of shaking-hands with a non-mahram who is non-Muslim?
Based on Shia perspective regarding shaking hands to non-mahram, this is not allowed to shake hands to non-mahram even though s/he is non-Muslim. Acutally there is no difference which religion s/he is from, haply it is considered as a haram practice which could be related to the probability of sin.
There is nothing wrong with shaking-hands with a non-mahram if they extend one to you. But there are limitations, i.e. the handshake should not be lascivious (rarely but mentioning it just in case) and shouldn't involve skin contact with a non-mahram.
The latter limitation needs elaboration. One cannot shake-hands, or have physical contact (generally speaking), with a non-mahram if that act involves skin contact.
But doesn't that negate the starting statement?. No, it does not. If one has gloves on, then they can handshake. If wearing full-sleeve shirt then extend the wrist (would look odd but better then declining). Or anything that doesn't involve skin contact (plus convenient). Else one shouldn't and could politely say it's against my convictions and apologies for not being able to.
The act of handshake has many positive aspects and is largely practiced in different cultures (happens to reminds me of Ace Ventura: When Nature Calls ;)) but it should be avoided if it goes against the Islamic principles.
AFAIU, this ruling would be consistent across majority of the Islamic school of beliefs.
PS: there is relaxation to the latter ruling (skin contact) in cases of life-and-death and so is the case with many other rulings.
You as a Muslim are very lucky to have your religion. And we live in a society where we have obligations and some of them are small such as shaking hands. So what you shake a hand of a non-Muslim, as long as you know we're you stand as a Muslim and your boundaries. There are a million haram distractions in the west. We as Muslims have to show ourselves as being polite curious and well mannered. Unlike some societies, were the believe we Muslims are the devils. What is in your heart in regards to your religion how strong is your fear of Allah is what counts. Shake your hand with a non-Muslim so what,
A handshake is a short ritual in which two people grasp one of each other's like hands, in most cases accompanied by a brief up and down movement of the grasped hands.
Using the right hand is generally considered proper etiquette. Customs surrounding handshakes are specific to cultures. Different cultures may be more or less likely to shake hands, or there may be different customs about how or when to shake hands. Handshakes are known to spread germs. So would suggest washing your hands before you eat.