Me and Bobby McGee - the incomparable Janis Joplin
Where Does the Good Go - Tegan and Sara
When You Say Nothing At All - Alison Krauss
Wishing You Were Somehow Here Again - Sara Brightman’s version especially
Strange Fruit - Billie Holiday
Gloomy Sunday - Billie Holiday
I Don’t Know How to Love Him - Yvonne Elliman
Someone To Watch Over Me - Ella Fitzgerald
^5
I almost listed this, but thought people would snork. I kinda saw the OP as asking for grrrrr songs, but I see now that in this case intense can encompass slower songs.
I will add all of the “Emily” songs from the Indigo Girls album Swamp Ophelia. “Least Complicated” nearly makes me cry after all these years, but then I identify strongly with it. Well, that and “Language or the Kiss.”
And if you want to REALLY tear up, try Dar Williams’s “When I Was a Boy.” Or is that just me?
Good choice for Kate, although I would have suggested Pull Out The Pin from The Dreaming, sung from the perspective of a Viet Cong. Or A Coral Room from Aerial, where she deals with her mother’s death.
That’s an awesome way to describe the way music can make a person feel. I like it.
Walkaway Joe - Trisha Yearwood – depressing, soulful sort of intense
But Fate’s got cards
that it don’t want to show
I’m not sure if it fits quite right, but I bawled when I first heard Sherrie Austin do “Streets of Heaven”.
You know how an intense song can bring you back to the place where you first heard it? It probably doesn’t fit into your scheme, but when I first heard the Dixie Chicks’ “Taking the Long Way” I was driving with a friend on a lonely highway and it just seemed, I don’t know, right.
Yes, I am a music troglodyte who needs to get out more.
Shoot, forgot one:
Rain – Patty Griffin
LUCINDA WILLIAMS, pretty much anything by her is phenomenally intense, and I see a lot of Nashville copycat by the younger gals lately.
I tried to list favorites, but all of her songs are intense and amazing. Just check her out. The most accessable and popular would be her album “Car Wheels on a Gravel Road”. Her newer work is is more sublime, but intense on a different level. Lucinda paved the way for modern Alt-country singer songwriters.
I have every single one of those songs in my collection, either on vinyl or CD. Wow–a musical twin!
Tori Amos: “Me and a Gun” and “Winter” (hell, pretty much everything from “Little Earthquakes.”)
I see your R.E.S.P.E.C.T. and raise you Until You Come Back To Me (That’s What I’m Gonna Do).
(Nowhere near as good as the original recorded version, but it does have Stevie Wonder (one of the writers of the song) on piano and trading verses with The Queen of Soul!)
CMC fnord!
BTW folks GREAT list of songs that belong in every collection. 
k.d. lang - Pulling Back the Reins (Not a torch singer. A frakking flamethrower singer.)
Sinead O’Connor - This is the Last Day of our Aquaintance
Angelique Kidjo - the album Aye (It has been a while since I listened to this album, so I don’t know what to recommend).
Linda Ronstadt - the albums Canciones de mi Padre and Mas Canciones (ditto on both counts)
Other people know Hungry Lucy? I’ll be damned.
Somebody Hates You by Snake River Conspiracy
Front Row by Toby Lightman
Both Hands by Ani DiFranco
You Promised Me a Symphony by The Gathering
Cowboy Junkies - Tons of songs, but I’ll pick “Spiral Down” and “Misguided Angel”.
“Wait for an Answer” - Heart
“Missed Me” - The Dresden Dolls
Love them and have multiple albums. They’re amazing.
Chain Gang by The Pretenders (literally raises the hair on my arms during the bridge section of the song, with those soaring harmonies)
Selling Jesus by Skunk Anansie
Human Behavior by Bjork
Most Karen Carpenter stuff, for some reason.
Ray Of Light by Madonna
This is EXACTLY what I was coming in to post. As for a particular song, off the top of my head I’d suggest Those Three Days.
The entire album Haunted, by Poe did that to me. It’s still a go-to when I’m furious or hurting.
In particular, I think Control and Wild do that.
I also think Melissa Etheridge’s songs Breakdown and The Different fall into the category of “intense” for me.
Marianne Faithfull ** The Ballad of Lucy Jordan** Song about a woman who slowly goes nuts in her white suburban bedroom in a white suburban town.
Emmylou Harris Red Dirt Girl, Bang the Drum Slowly
Ball & Chain - Janis Joplin
Hound Dog - Big Mama Thornton
At Last - Etta James
St. Louis Blues - Bessie Smith