My favorite cutscenes are when Zelda reveals to Link that she is the goddess reincarnated...that whole scene. It was deep and it made me feel for Zelda in a way.
I also liked the scene where Demise was resurrected and the following scenes with him as well, especially the final scene with him after you defeat him, promising a never-ending curse on Link's and Zelda's descendants, and telling Link he (or his hate, whatever) would be reborn again and again.
I love when Link first meets Fi and you get the Skyward Sword. It gets me so psyched up and I love Fi's theme playing there.
Opinions on Fi because banner: I really like Fi for her character design but the way she is used in the game is, IMO, part of a direction Nintendo has been going since WW that I disapprove of: having the game play the game for you.