I definitely have. Some songs either have special meaning for me, or they're just really good. Like, I'm not super-religious, but two of my favorite songs are "On Eagle's Wings" and "Ave Maria," the former because it was sung at my great-grandfather's funeral some years back and was just done so amazingly beautifully, and the latter because it was sung at my parents' wedding, also amazingly beautifully (at least half of the guests were crying by the time the lady finished; it was probably the most gorgeous thing I've ever heard).
On the so-good-that-I-just-react note is the song "It Isn't That" (really, 그게 아니고, "Geu-ge anigo"), by 10cm. The lyrics tell about how this guy is reminded by all these little things of a girl that left him, but he insists, "It isn't that; that's not why I'm crying," and his voice is just so powerful, and I can't say enough good things about the song >.> [link]
The Queen and the Soldier by Suzanne Vega is the most emotionally haunting song I've ever heard. Other than that, it's mostly songs from musicals that can make me cry. I suppose I need a story to go with it, LOL!