What moments of wonderful silliness & unexplained magic in cinema really stand out in movies for you? I don’t mean spectacular special effects like Gandalf’s battle against the Balrog; I mean mostly-realistic movies that suddenly veer in strange and wonderful territory?
I’ll start off by nominating the "Chori Chori scene from the Guru. I’m sure someone else will find a better one quickly, though.
I like the musical interludes in The Happiness of the Katakuris
It’s a film about a guy who buys a hotel in the middle of nowhere because he thinks a new road is going to be put through there. He sinks his life savings into the venture and moves in with his family to wait for the new road to make them rich.
Not only does it not happen but the few guests they have all die in various unexpected ways. For example, two guests are clearly having a dirty weekend. One is built like a sumo wrestler, the other a very slight young girl. The guy has a heart attack mid-act and the girl is crushed to death by his corpse. Umm, it’s funnier than I just made it seem.
Anyways, for no reason I can remember, at several points in the movie there will be a cut to a song and dance routine by all the guests who have so far died. And IIRC the ones that died first will be more decomposed than the later ones.
I’m going to put this on my rental list now 
Lots of little things in Amelie, but especially the pictures in her room moving.
Not a movie, but any time they break into choreographed song in the “Singing Detective” series
Not really “silly,” but…
This scene, from Magnolia.
You know the scene I’m talking about.
Possibly one of the most polarizing moments in recent cinema. Those who don’t like the movie *loathe * this scene, but to me, it’s the moment the film transcended itself, reaching something resembling greatness.
I was already loving the movie before this bit, but there’s something about the ‘lighting of the beacons’ sequence in Return of the King that really moves me. It’s just so beautifully shot.
The Grand Central Station Waltz from The Fisher King.
From “Garden State”:
The party scene
The alligator skating scene
The incomprehensible song-and-drinking-game from Shanghai Noon. 
Link, man, link!
I’m not sure if this counts as what the OP is looking for (since it definitely was in no way “silly”) but in *Schindler’s List *there’s a little girl in a red coat who wanders among large crowds of people in one or two scenes. Her coat is the only bit of color in a movie that’s otherwise entirely in black and white, and it’s a very powerful effect.
There’s a series of scenes in Shakespeare in Love that have a totally dreamlike, transcendant quality. It’s just after the lovers discover each other, and scenes flip back and forth between lovemaking and rehearsals, and how the two things feed each other. It’s all tied together by the score. The whole thing lasts about seven minutes.