It is permissible for a wife to shake hands with her father in-law because as soon as a man conducts a marriage contract with a woman, his father becomes a Mahram for her.

The wife’s sister is not a Mahram to her sister’s husband, so it is not permissible for him to look at her, or be alone with her, or to shake hands with her.

The husband of a woman’s daughter is one of the Mahrams through marriage, so it is permissible for him to see of her what he is permitted to see of his mother, sister, daughter and all other Mahrams.