“Gone Away” by The Offspring
Another good one is “Healing Time” by Poor Katrina. I know, you’ve never heard of them. Unless you went to UNH in the late 90s, there’s no reason you should have. I used to work with the male singer 
I’m not going to leave it up long(so don’t look for it months from now if you stumble across this thread), but here’s a link to it.
How did I not think of the Rolling Stones’ “Angie”?
“Angie, I still love you, baby. Everywhere I look, I see your eyes. There’s not another woman that comes close to you… c’mon, baby, dry your eyes…”
I dunno, here is a Firefly tribute video that uses the song. It’s wistful. Also, the OP didn’t say “bands whose songs are often sad, macabre, strange and depressing are disqualified.” either. 
This is a great song, and Johnny Cash is one of my favorite artists. I only wish he had recorded this song 5 or 10 years earlier, when his voice was stronger and more expressive.
The frailty you can hear in his performance adds depth to the song I think. He might not have been able to put the same strength into the performance that he once could, but he still got the meaning across. It was a sort of opus by him. All his life’s experience went into that performance, all that he had felt, and was feeling.
The OP did say “-est” and “ever”, but realizes now how impossible that is. 
But sticking to love songs should be no problem.
For example, also by Johnny Cash who I love, "Give My Love to Rose". It’s about love, but not about lost love.
Johnny had a most excellent TV show in '69 or so. It was, I think, the best music show on TV at the time, but he pissed off a lot of his “core” audience.
yup. I came here to mention this one.
This is the total redneck Romeo and Juliet.
The line bolded always touches a note with me.
Grace, all the more because it’s true:
As we gather in the chapel here in old Kilmainham Jail
I think about these past few weeks, oh will they say we’ve failed?
From our school days they have told us we must yearn for liberty
Yet all I want in this dark place is to have you here with me
Fields of Athenry is up there, too.
Two Pearl Jam songs come to mind:
Of course, there is always Terry Jackson’s Seasons in the Sun, the Verve Pipe’s The Freshman, and Can’t Live by various artists…
Grapefruit moon - Tom Waits!!
lyrics!
Grapefruit moon, one star shining,
shining down on me.
Heard that tune, and now I’m pining,
honey, can’t you see?
'Cause every time I hear that melody,
well, something breaks inside,
And the grapefruit moon, one star shining,
can’t turn back the tide.
I love the Indigo Girls version of “Romeo and Juliet” even more than the Dire Straits original.
Here are the two saddest love songs of all time…
“Lover, You Should’ve Come Over” by Jeff Buckley
And the most heartbreak in the fewest lyrics…Josh Ritter’s “Drive Away”.
Lyrics don’t mean much when its a song about feeling…and true sadness is all feeling.
“My Immortal” doesn’t have tragic lyrics but it’s melancholy is powerful.
Cheap Trick’s The Flame.
Righteous Brothers’ Unchained Melody.
Another vote for “He stopped loving her today.”
Also the most unheralded soul group in history “The Checkmates Ltd.” singing “Love is all I have to give.”
No love for Johnny Preston’s Running Bear ? Yeah, I guess it was kind of sappy.
More seriously, I nominate Dylan’s Simple Twist of Fate .
“Independence Day” by Ani DiFranco
Another vote for “Romeo & Juliet”, just to hammer home that it wins the thread 
It is widely regarded as being about Knopfler’s romance with Holly Vincent, an (ultimately) minor recording artist herself.
If you want a real rollercoaster ride, listen to “Romeo & Juliet”, then “It Never Rains”, then “Love over Gold”, the latter two being tracks off Dire Straits’ Love Over Gold set.
Other band members have confirmed that their understanding is that all three are about Holly Vincent, though I’m not sure how much they know and how much they are guessing given Knopfler’s legendary reticence (his magical ability with the written word and his mumbling ineloquence in any interview I’ve ever heard are remarkable for their disparity).
“Romeo & Juliet” is of course all about a starry eyed lover singing in stunned disbelief, which kicks me in the guts every time. “It Never Rains” is an angry, vicious and slightly gloating song accusing an ex-lover of whoring herself to further her career but ultimately getting nowhere - “you threw people over on your way up, 'cos you thought that you were never coming down”. Finally, “Love over Gold” is more forgiving - picturing a former lover as being more capricious than deliberately hurtful - “you go dancing through doorways, just to see what you will find…”
Gosh, I think I’d better go down to my workshop and do some manful things: I’m coming over all mushy just thinking about this stuff.
I keep thinking of more…
While this is somewhat untraditional, I’ve always been extremely moved by Aerosmith’s ‘‘What it Takes,’’ which I hold as their best ballad to date.
*
Well I used to feel your fire
But now it’s cold inside
And you’re back on the street
Like you didn’t miss a beat*
Garth Brooks has some particularly sniff-worthy songs, too. The one that always rips me up is ‘‘Learning to Live Again,’’ which as far as I can make out is a man who lost his wife in a car accident and is trying to re-enter the dating scene at the urging of his friends.
Little cafe, table for four
But there’s just conversation for three
I like the way she let me get the door
I wonder what she thinks of me.
At the end of the date, he realizes he really does like the woman, and he asks if he can see her again.
And she just smiles her best smile
And she laughs like it’s going out of style
Looks into my eyes and says, "We’ll see.
Oh this learning to live again is killing me.
God this learning to live again is killing me.’’
This song is on my “Music to Commit Suicide By” CD…