This is a tough category- mainly because couples are so weird. Even screen couples have issues. There is no way to pick them on a consistent thread - we'll just have to go with a criteria of interesting, loving, great clothes and a sense of humor and settle for a mostly happy Hollywood ending.
"The Thin Man" (1934) - One of the best screen couples- Charming, sophisticated, in-love. While there are smidges of Nora being a silly woman and moments of Nick's superior attitude, it is part of the characterization- overall they behave like a partnership and wear the most fabulous clothes.
"Now Voyager" (1942) - Oh its just delicious, Claude Rains, Bette Davis and Paul Hendreid - just watching them is enough. The story is a bit contrived and over the top, but the final line of the film is so perfect, filmed with qualities that make a scene captivating and unforgettable. (You'll just have to watch it and see)
"The Ghost and Mrs. Muir" (1947) - Usually I object to films that don't allow the couple to be together. What is it with Rex Harrison and his almost love affairs? There are so many good elements to this film, their conversations, clothes, the setting, its a really terrific story.
"My Fair Lady" ( 1964) - Ah, what did I say about Rex Harrison and his almost love affairs? It would have been so much more satisfying to have Professor Higgins sweep Eliza up and carry her off.....but that would have been a different movie. While not exactly 'romantic' this couple is perfect for each other.
If I were voting- it would be Nick and Nora Charles- a fun and interesting relationship.