It’s also known as Stairway to Heaven, and yeah, it’s a great movie! David Niven and Raymond Massey are in top form.
Another musical: An American in Paris.
Seconded for Holiday, with Cary Grant and Katharine Hepburn. It’s one of my favourites. Funny, poignant and beautifully written.
I’d also suggest Marty, with Ernest Borgnine and Betsy Blair. Fabulous film.
I love Holiday, but it’s a little headier than some of the other movies. It’s not a screwball comedy, like Ninotchka or Bringing up Baby, and it’s not a pure love story, like Sabrina.
If you are picking only five, I’d go for the lighter fare, albeit, if I had to rank the films listed so far,* Holiday* would be very close to the top.
The Awful Truth
Top Hat
City Lights is a wonderful (if ultimately heartbreaking) love story.
The Awful Truth is on my top 10 list of movies, but I don’t know if it’s the sort of love story the OP is looking for.
I love David Lean’s Summertime, with Katharine Hepburn and Rossano Brazzi. It’s not only a good love story, it has awesome footage of Venice. And it’s David Lean! Which means its well directed and beautifully filmed.
It’ll make you want to take a trip to Venice.
IMHO the best romantic comedies have great chemistry between the lead and real-life chemistry is the best.
May I recommend Adam’s Rib?
Oh heck, I love almost everything Hepburn (with the exception of The Philadelphia Story).
So, add my votes for Summertime and *Bringing up Baby *and add in The African Queen.
How could I forget The African Queen? And it has a happy ending, too.
More recently, the 2005 Pride & Prejudice has a great cast, Oscar-nominated music, beautiful cinematography and a winning romance.
The 1995 mini-series is better.
If we can nominate post-1960 movies, let me throw in 1985’s A Room with a View. Beautiful movie.
I second “Marty”, one of the most realistic portrayals of romance ever put on film. These aren’t “pretty” people. They’re real. I could watch this movie over and over.
I haven’t seen it mentioned (may have missed it) but my personal favorite is “Roman Holiday”
Okay, it doesn’t have a happy ending, but it has the only ending it really could have. Plus Audrey Hepburn is drop dead beautiful, Gregory Peck is great and Eddie Albert brings comic relief.
And while this film may not be considered a romantic film, per se, the main plot of “The Apartment” is built around an unrequited romance between Jack Lemmon and Shirley Maclaine and it’s a very funny film (with a happy, funny ending)
Another film by Paddy Chayefsky: A Catered Affair. Ernest Borgnine (again) and Bette Davis.
The TV version of Marty was also very well done: Rod Steiger, Nancy Marchand, Betsy Palmer. Plus we find out what happened to Major Hochstetter after the War. 
The ultimate romantic comedy: Some Like It Hot.
Even though I generally don’t care for chick flicks, I quite liked Sense and Sensibility with Emma Thompson and Kate Winslet (and Hugh Grant, and Alan Rickman). When I was teaching English as a foreign language, I’d show it to my students every 8 March (International Women’s Day).
Are you referring to the pairing of Tony Curtis and Marilyn Monroe, or to that of Jack Lemmon and Joe E. Brown?
The movie wouldn’t be half as funny without the latter.
If you want variety, I would suggest spitting the lists above into genres. Pick one from each list:
A) Classic screwball comedy
(ie It Happened One Night, Bringing up Baby, His Girl Friday, too many others to name but listed above)
B) Tearjerker Romance
(ie Waterloo Bridge, Brief Encounter, City Lights)
C) Sweeping Technicolor Romance
(ie, The African Queen, Summertime)
D) Musical
(ie An American in Paris, Top Hat, Easter Parade)
E) Drama/Other Romance
(Marty, To Have and Have Not, Sabrina, Roman Holiday)
An outlier that I really like: I Was a Male War Bride, with Cary Grant and Ann Sheridan.
The OP asked about movies, and anyway, you’re wrong.