I'm all for songs that speak to my heart.
My musical tastes are all over the map, though. I think I can find a favorite in almost every genre (*including* Trip Hop!)
I love sad, mournful songs (and yep, Mad World (Jules version) is in my frequent playlist).
On a different note (heh - see how punny she is), I also love classical music. You can get me going with sweeping opera choruses and opera duets, no problem. "The Flower Duet" from Lakme is a favorite as is Barber's "Adagio for Strings." Man. Nothing tears up my heart more than a deep, mournful cello piece (ala "Mad World" or "Eleanor Rigby") or a series of strings in a sad piece, ala "Adagio" or the "Schindler's List" theme.
Other unusual pieces I like include:
Patsy Cline's "Sweet Dreams (of You)"
U2's "One"
REM's "Losing my Religion"
The Smith's "Meat is Murder"
Loreena McKennit's "La Serenissima"
Chasing Furies' "Fair Night's Longing"
Frou Frou's "Let Go"
Dire Straits - "Brothers in Arms"
The Dandy Warhols "Sleep"
Dryve "Rain"
Leeland "Tears of the Saints"
Leonard Cohen's "Hallelujah"
New Order's "True Faith"
Saybia's "Eyes on the Highway"
Oren Lavie's "Her Morning Elegance"
Sting's "A Thousand Years"
And just about anything by Gustavo Santaolalla, Coldplay, Elliott Smith, Explosions in the Sky, Pete Yorn, Radiohead...etc.