The Jazz Singer (1926)
Okay, so this is definitely not a movie that has aged well. In fact today it might be considered outright racist due to the infamous scenes of the main character performing in blackface. However, it did make a significant impact due to its technical accomplishments. The Jazz Singer was, at the time of its release, a huge gamble for Warner Bros. Studios. Sound technology had been experimented with for decades, but this is often credited as the first feature-length sound film (though technically it isn't so much a talking picture as it is a silent movie with short segments incorporating sound, usually for musical numbers). It was this film that brought sound to the mainstream and made studios realize just how profitable talking pictures could be.
The Red Shoes (1948)
If one really wants to get technical, this film is mainly about the struggles of various characters to navigate the ballet industry, but ballet, by its very nature, based on music. The plot also centers around a performance of the ballet. As a result, it involves some incredible musical sequences, including an amazing (if heavily condensed) performance that actually managed to make me respect ballet for the first time in my life. We also can't forget that one of the other main characters is a musician and a composer, and his musical career is an integral part of the story.
Walk Hard: The Dewey Cox Story (2007)
This bizarre film is a hilarious satire on the life of rock and roll icon Johnny Cash, represented here by the fictional character of Dewey Cox. The story of the film naturally concerns the strange events contributing to the rise and fall of Cox as a musician, who struggles with some unusual situations. Along the way, there are also cameos by famous rock stars. To call this movie surreal is an understatement, but it is still a lot of fun and definitely worth watching.