The Thin Man series (1934-1947) - Nick and Nora Charles
This is the original fun couple, the one that all others would be indebted to. Playful, flirtatious, witty beyond normal human capacity and a little wicked, these two love martinis, mysteries, and of course, each other. I say "wicked" due to classic moments such as Nick throwing Nora into a taxi and telling the driver "Grant's Tomb!" to get her out of the way.
The Potters have supporting roles, but are no less memorable. Edward Everett Horton plays Nick in both this version and the one from 1931, so naturally he's a much older friend of the main character in the second outing but he still gets a sassy wife. He and the missus have the kind of raised-eyebrow rapport that can only lead me to one logical conclusion: THEY HAVE A TON OF SEX. I doubt anyone else reached this verdict, but I really felt like they were doing it whenever they were offscreen. And that's awesome.
Heaven Can Wait (1943) - Martha and Henry Van Cleve
This movie is rather unusual in that there isn't really a conflict or villain, it's more just the story of a man's life, the good and the bad. In not trying to force a plot, the audience gets the treat of seeing a marital relationship unfold very naturally over the course of 40-odd years, from lusty youth when Henry steals Martha away from his loser cousin to sweet and gentle old age. And to the afterlife and beyond...(no spoilers here - come on, look at the title).
The Best Years of Our Lives (1946) - Milly and Al Stephenson
This postwar readjustment drama still holds up today, largely due to the genuine performances of Myrna Loy, Harold Russell and Fredric March (the latter two snagged Oscars). Loy and March play a couple readjusting to married life in one of the more refreshingly realistic depictions of a long-term relationship from this era or any other. When their teenage daughter accuses them of never having relationship problems, Milly sighs and says there were countless times when they hated each other. "How many times have we had to fall in love all over again?"
Adam's Rib (1949) - Adam and Amanda Bonner
Two married laywers on opposite sides of a criminal case - conflict of interest much? Many agree that this is the best of the Hepburn and Tracy collaborations, and while I haven't seen them all, I'm not sure what else you could ask for. It's a battle of the sexes through and through, but by an unforgettable ending involving licorice you'll know that these two really love each other. (I didn't mean for the licorice thing to sound dirty - I promise it's really not).
Spencer Tracy puts his grump persona to great use in this film as the frazzled title character, and Joan Bennett as his wife knows just how to handle him. For all their ribbing, though, the moment when Stanley sees his wife in her mother of the bride gown is really sweet and touching. And they share a slow dance after the sanity-sucking craziness of the wedding is over. It just shows that you can still be husband and wife even after you're mother and father (you know, sometimes).
Okay, I'm slightly cheating, because they're not married for the WHOLE movie (their courtship and marriage comes in the first third or so) but it's still not a final-reunion-in-the-rain-at-the-end-of-the-movie situation. At first you kind of wonder what the shy Norma sees in the belligerent, awkward, and inarticulate Rocky, but as their marriage progresses you figure it out (and so does she). She's the only one who can talk sense into him (he beats guys up for saying much less), and when Rocky comes home in a fit of rage after losing a fight, she puts a lamp right of front of him when he starts to swing and we realize they're a match made in heaven.
Not that much of a stretch, since Woody Allen was directly and visibly inspired by the Thin Man films - it's Nick and Nora, but 20 years later, neurotic and Jewish. Allen and Diane Keaton are reunited after his Mia Farrow period, and despite the Thin Man influence there's the classic Allen touch, with lines like "Save a little craziness for menopause!"
While most of their time is spent having Homer drive Marge crazy, there's a love there unlike most other relationships in pop culture - and in being so, is actually quite like most relationships in real life. Maybe it's partially because it's had 20 years to develop, but the writers have really nailed the decidedly unglamorous but still sweet dynamic between average American spouses. Homer always manages to prove his devotion in the nick of time, and Marge remembers why she keeps him around. (On a somewhat related note, a Google image search for "Marge Homer" reveals SO MUCH DISTURBING SIMPSONS PORN THAT WILL HAUNT MY DREAMS.)
And a controversial bonus pick:
Who's Afraid of Virginia Woolf? (1966) - George and Martha
I first read this play and saw the film in an American lit class, where the professor introduced it as a love story. The first time you see it, all you can do is be shocked at how these people treat each other. That's fine. Get that out of your system. But I saw it a second time and realized that my professor was right. There is no one who could possibly be with George or Martha but the other. The games they play with each other, although seemingly unhealthy, are what sustain them. It's like how sadism looks painful and weird to most people but is a source of fun and intimacy for some. If you can get past the fact that it's two people screaming at each other and you were taught that two people screaming at each other typically indicates serious problems, then perhaps you'll see the romance.
This list is by no means exhaustive...who did I miss?