I'm more into music scores then "songs", but here are a few music/film combinations I like.
I love the drums and Yo-Yo Ma's cello in Crouching Tiger Hidden Dragon. We all know those scenes.
I love the classical piece during Akira Kurosawa's hellscape battle in RAN when the great Lord fully realizes the futility of his power.
I love the Noh scene in Banshun when the daughter realizes her post-war father/daughter life is about to come to an end, because the father is ready to get married again, which means she too must marry and leave his house. When I hear Noh I think of this movie. It's one of the best wordless scenes on film in my opinion (but I'm biased because of my crush on Setsuko Hara).
I love the piano scene in Touch of Evil with Orson Welles and Marlene Dietrich. "You a mess honey" Whenever I think of that movie I think of that piano.
The guitar audition from Inside Llewyn Davis is heartbreaking, in the end, because the audience cannot comprehend what the artist is trying to reveal.
I MUST sing along to the Tiny Dancer scene in the movie Almost Famous. It's impossible to resist.
The tango scene in Scent of a Woman is fantastic. "Some people live a lifetime in a minute"
And pretty well every song in Phantom of the Paradise.