First of all, there is the scene where Victoria first meets Victor. Victor awkwardly steps into Victoria's home and seeing a piano, starts playing a beautiful piano piece. I like this scene not only because of the music, but because it shows a lot about Victor's character. See for yourself!
I also like the scene where, after Victor tricks Emily into letting him go back to the "living world", he tries to make things up for Emily. Emily's inability to stay angry at Victor show how she feels about him. I think it's a genius way of showing what it going on between the two characters without having to say anything.
Another song, called "The Remains of the Day" is sung by the dead and narrates Emily's tragic story. The funky and catchy music as well as the underlying story in the lyrics makes the music a major part of the movie. In addition, the visuals also show what the Land of the Dead is like.