At the beginning of the film British fashion icon Lady Holiday (played by Diana Rigg, of "The Avengers" TV show fame) is robbed of some jewels. Kermit, Fozzie and Gonzo convince their editor to send them to London in order to cover the case and solve the crime.
Once they arrive in London the boys rent a room at the Happiness Hotel, a run-down old ramshackle that's home to tons of characters from The Muppet Show.
Lady Holiday & Nicky
Around this time Miss Piggy makes her first appearance in the film. She is introduced at Lady Holiday's office, where she applies for the job of "high fashion model." Instead, Lady Holiday gives her a job as her secretary. When Kermit goes to Lady Holiday's office to get an interview, he sees Miss Piggy and mistakes her for Lady Holiday. Miss Piggy is instantly attracted to Kermit and she allows him to continue believing she is the really Lady Holiday.
Nicky & Piggy
At the same time Lady Holiday's no-good brother Nicky meets and falls in love with Miss Piggy. The problem here is Nicky is the one who stole his sister's jewels and, along with his team of lady models, he plans to steal even more of his sister's possessions.
Later, Kermit, Fozzie, Gonzo and Miss Piggy are having a date at a classy restaurant (well, more of a supper club). The real Lady Holiday is also there, along with Nicky. When Lady Holiday is robbed of a diamond-covered necklace, Miss Piggy's charade is over, and Kermit learns she is not really Lady Holiday.
"All the world ever wanted was you!"
Later at a fashion show, Nicky plants the stolen necklace on Miss Piggy and, in front of everyone at the show, he frames her for the crime. Miss Piggy is taken to jail. In order to clear her name, Kermit and the other Muppets decide to catch the real jewel thieves.
When Nicky and his cohorts try to steal Lady Holiday's fabulous Baseball Diamond, the Muppets are there waiting for them. By this time Miss Piggy has escaped from prison and she rides in on a motorcycle to help the other Muppets capture Nicky.
The film ends with Miss Piggy clearing her name, Nicky and his team going to jail, and Kermit, Fozzie and Gonzo getting their story.
Breaking the Fourth Wall
Gonzo the photographer
One of the weirder things about The Great Muppet Caper is that the Muppets and some of the cameo guest stars know they're in a movie. From the very beginning, Kermit talks straight to the camera and explains to the audience that they are watching a movie, and who plays what part.
A later scene in the film has Kermit calling out Miss Piggy for overacting and "hamming it up." They both get upset, and then quickly calm down so they can finish the scene and get on with the movie.
A very brief cameo
Near the end of the film, Oscar the Grouch and Peter Ustinov appear in a cameo scene together. When Ustinov asks Oscar "What are you doing here?", Oscar replies "A very brief cameo." To this a defeated Ustinov answers "Me too."