Singing in the Rain (1952)
Okay, this does have Gene Kelly in it, but it is the ultimate Gene Kelly musical! If I could only watch one Kelly movie for the rest of my (the horror!), it would be this one. Debbie Reynolds is beautiful in it. And it is so funny! Wonderful songs, wonderful dancing. Love it.
R&H's Cinderella (1957)
This is the very first R&H Cinderella. History lesson: This was written for TV, and this was a live broadcast--there was no dubbing, no going back and cutting. And I love Julie Andrews. She is beautiful, graceful, classy, and had a wonderful voice. The clip above is "My Own Little Corner," but if you watch all of it, you will see that her ball gown is far superior to any other R&H Cinderella, as is the rest of the musical. Unless, you know, you like the impossibility of the Brandy version, with a white king, a black queen, and an Asian prince.
My Fair Lady (1964)
Who doesn't love this musical? I do have to say, that I think Julie Andrews was cheated out of the part of Eliza, considering she played her on Broadway, but I do have a love for Audrey Hepburn. And Marni Nixon (who dubbed Audrey's singing) has a wonderful voice (you have also heard her voice in other movie musicals such as The King and I, and West Side Story).
The Sound of Music (1965)
Another Julie Andrews. What can I say? I just really like her. This is another movie that I think it is impossible not to love. Christopher Plummer is so handsome and has those beautiful bright blue eyes. The songs are wonderful, the scenery sublime, and the kids are just so darn cute. This is the ultimate love story.
Finian's Rainbow (1968)
This is a movie that most people don't know about, which is a shame. It has Fred Astaire in it (obviously an older Fred Astaire, not that that hampered his dancing at all), and Petula Clark. It is the story of an Irish man and his daughter "following the rainbow" to a small American town. The scene I linked to is one of my favorite songs, but it isn't a very good introduction to the movie, which I apologize for. Basically, just get the movie and watch it. It's very 70s, but has great music and a fun story (including leprechauns!)
Fiddler on the Roof (1971)
This scene was my favorite when I was little. I used to rewind the tape and watch it over and over again, driving my siblings crazy! This is a heartwarming, sometimes funny sometimes sad, story with beautiful music.
Slipper and the Rose (1976)
This is another not very well known musical, but one that I grew up watching. It is Cinderella, but my favorite thing about this movie is that it doesn't end with finding Cinderella, it goes beyond that to the difficulty of the prince wanting to marry a servant girl. You will laugh at the King and the Dowager Queen, you will sigh with satisfaction when Cinderella and the prince first meet, and you will be singing the beautiful songs. My favorite song is "Tell Him," sung by Cinderella. It's a wonderful version of the Cinderella tale. (BTW, the costumes are gorgeous!)
Moulin Rouge (2001)
Let's skip several decades and get to the 2000s. Moulin Rouge. It's a slightly risque film, as well as being bizarre, but the integration of contemporary songs is great, and the love story is beautiful. The song I linked to is "Roxanne." Basically I just love that they made Roxanne a tango.
The reason I linked to this scene in Hairspray is because I think the words are so hilarious! "Nice WHITE kids," and "He's Corny!" Hairspray is fun and colorful, with catchy music, and a great cast.
Okay, Joss Whedon is a genius. I love his work and this is no exception. This is the story of super villain, Dr. Horrible, the girl he loves, Penny, and his arch nemesis, Captain Hammer. It's completely ridiculous, but you will love it. The song I linked to is a song that I used to sing every time I did laundry in my apartment's laundry room. "Underthings, tumbling."
Anyway, there you go. Hopefully you have been introduced to some movie musicals you haven't seen before. Go watch them! You won't regret it, I swear!