I don't really know many song titles. I tend to listen to full albums. But I guess I can list my favourite songs from my favourite albums.
The Antlers - Two. Probably my favourite song right now. It's catchy and the lyrics are very, very, very (very) depressing. Listen to it: http://www.youtube.com/watch?v=yAAD61ZruSE
The Flaming Lips - Race for the Prize. Very fast paced and fun song.
Mad Season - Wake Up. Probably Layne Staley's best performance. He's better in Mad Season than in Alice in Chains in my opinion.
The Antlers - Epilogue. The ending of the record. In context to the rest of the album this song is extremely depressing. I almost cried the first time I heard it. But you should listen to the whole album and pay attention to the lyrics if you want to enjoy this song.
Neutral Milk Hotel - Two Headed Boy Pt. 2. Also a depressing song but in a different way. The lyrics are different but good. I like it.