For me, The Smashing Pumpkins play most of my favourite Sad songs...
Here are my saddest...

Appels & Oranjes
("What if the sun refused to shine?, what if the clouds refused to rain? What if the silence let you dream? What if the air would let you breathe? What is, what is isn't you? What are you gonna do?")

Bjork songs also affect me...

Who Is It?

Dido...

Here With Me