Is there another group that performed "Creep", as the version in this video is not as "heavy" as the one I am used to hearing.

Oh, and Mad World has to be the most depressing song I've ever heard in my life. Thanks a lot. :(

https://www.youtube.com/watch?v=_7LCHwdd5nM

Parts in this song give the frisson.  I am bad at embedding videos, so its a link.

Are you thinking of this song?

Shame. You were allowed to use every Radiohead song EXCEPT Creep

It's just a Reddit joke, lol.

Ok, now I'll actually post some songs. 

You could really post nearly every Modest Mouse song from their first 3 ... Dramamine, Tundra and Desert, Ohio, Talking Shit About A Pretty Sunset, Make Everyone Happy/Mechanical Birds, Teeth Like God's Shoeshine, Heart Cooks Brain, Cowboy Dan, Trailer Trash, Truckers Atlas, Bankrupt on Selling,  Styrofoam Boots, 3rd Planet, A Different City, The Cold Part, Alone Down There, The Stars are Projects, A Wild Pack of Family Dogs and Lives. But THIS is probably the best one.

Pretty much everything on this entire album, to be honest. 

I think if we're talking about the classic definition of songs that give you chills, then these 3 are probably the most classic examples in history: 

This is a pretty poppy example but a very good one, too: 

And if we're talking about songs that give you chills because of the movie they're in, then this is a prime candidate: 

Although honestly, I'm pretty sure the actual song is more chilling than the entirety of Silence of the Lambs .... 

If you want to see a chilling example of possibly the most depressing mindspace ever, that truly makes you think about humanity in the deepest of ways, this is a good choice:

And of course, you can't make a list of songs that give the chills without post-punk!!! 

It's very arguable that no song can communicate what this song tries to communicate, as well as it. 

If you want to meme and cry at the same time: 

And of course, TRUE David Bowie fans know that his most chilling song is not the Man Who Sold the World, but The Bewlay Brothers!:

Okay but seriously, last of all, if we're talking about chills given from nostalgia and the passing of time, then this is a good pick: 

It's a happy song, and I barely heard it as a kid, but it definitely seemed like it was in every commercial back in the mid 2000's (and probably some shitty youtube vids too)

Also OP, Jeff Buckley is an excellent choice. You should listen to his dad's music:

This is somewhat chilling, probably less so than the Studio version but still.

I'll be the weeb. Opening was good enough, but the full song is just that much better with frisson.

I like that song as well, but as it turns out, it must be the live recording that I must have recalled from memory. There's the two quick guitar power chords (I think they are called, I dunno) that are played after the "think you are special" part.

This is the uncontested winner, for me, in pure *frisson* power. Though, you'd probably need a bit more of intro than the couple seconds I linked to fully appreciate the climax. Maybe start at 35 min, or something, if you have the time.