Who are the men one is permitted to shake hands with, and is it permissible to shake hands with a sister's husband or an aunt's husband?

It is permissible for a woman to shake hands with her mahrams (unmarriageable kin) whom she is permanently forbidden to marry. These include: her son, brother, paternal uncle, maternal uncle, paternal grandfather (and higher ascendants), maternal grandfather (and higher ascendants), nephew (son of brother), niece (daughter of sister), and their equivalents through fosterage, as well as her stepson, her mother's husband (if he has consummated the marriage with her mother), her daughter's husband, and her husband's father.

This is based on the Almighty's saying: "Forbidden to you are your mothers, your daughters, your sisters, your paternal aunts, your maternal aunts, your brother’s daughters, your sister’s daughters, your foster-mothers, your foster-sisters, your wives’ mothers, your stepdaughters under your guardianship (born) of your wives with whom you have consummated marriage—but if you have not consummated marriage with them, then there is no sin upon you—and the wives of your sons who are from your own loins."

And also based on the Prophet's (peace be upon him) saying: "What is forbidden by lineage is also forbidden by fosterage."

However, the husband of a sister and the husband of a paternal aunt are not mahrams, and it is not permissible for a woman to shake hands with them, because they are not permanently forbidden to marry her.