One of the tropes (is it a trope, isn’t it?) That I love most for a villain is when we see his/her other face weaker and more sensitive.
Example: That the villain, at the beginning of the story, shows him/herself as a powerful, wise, self-centered character, very sure of him/herself and elegant to the public and the spectator. Clearly, a perfect person without weaknesses and without fears! :)
But then, history allows us to see in a scene or more the other side of the villain: As an insecure, aggressive, spoiled person, having a horrible attack of anger that nobody else sees in public, perhaps even showing his/her sensitive or mentally damaged side !
Yes, I am taking the example of Rock and Rule in a certain scene with his villain, but if you wish, you can share another example of a villain that you like very much!