One of Radiohead's more melodramatic tunes is "Exit Music (For a Film)" from the album OK Computer (please pardon any political advertising on YouTube...though the fact that "CommunistNihilist" has political ads attached to his/her video is very funny to me). This song lends itself to "classical" analysis, and is great for the end of first-year theory. It was supposedly "inspired" by Chopin's Prelude No. 4, though to my ears it's not all that similar besides the sol-le-sol element of the tune and the big sussy appoggiatura ending. The mood isn't the same, either. The Wikipedia entry for this tune suggests Electric Light Orchestra's instrumental "After All" as an influence; there are certainly melodic similarities, and both "Exit Music" and "After All" feature the move from a minor tonic to parallel major.
And while we're talking trivia, this is the Chopin Prelude that Jack Nicholson plays in Five Easy Pieces.
For music folks reading this (especially theory teachers), "Exit Music" has great examples of modal mixture, a secondary dominant, a Neopolitan (in function, if not an "N6", and I also like how the A minor chord foreshadows it), very clear non-chord tones in the vocal melody, and Picardy thirds (a sort of modal mixture, really).
I'd also like to mention that the first time I heard the fuzz bass come in I totally freaked out.
Sometimes it's foolish to try to analyze this sort of music in this way. But not this time. Here's my one-page kinda rough reduction and analysis (sorry that the figured bass is ugly).