'Howl' - Florence + The Machine
'Salt Skin' - Ellie Goulding
'Numb' - Marina And The Diamonds
'Toxic' - Britney Spears
'Tragedy (Austin Cello Version)' - Brandi Carlile
'Holding Out for a Hero' - Frou Frou
'Speeding Cars' - Imogen Heap
'In This Shirt' - The Irrepressibles
'The I Love You Song' - Original Broadway Cast of The 25th Annual Putnam County Spelling Bee
'The Con' - Tegan and Sara

(I CHEATED. THATS 10 SONGS.) (In no particular order)
Common threads: Uniquely textured/emotionally charged vocals, rich layering, nuanced mixing, unusual instrumentation, ...perhaps a sense of desperation? I'm startled to realize a lot of my favorite songs express profound emotional pain.