"If you scratch him, does his leg not shake?" Brian's court speech recalls Shylock's "if you prick us, do we not bleed?" speech in The Merchant of Venice.
The scene in which Lois knocks on the bathroom door and Brian says "Come." is a reference to Star Trek: The Next Generation. Captain Jean-Luc Picard does the same when someone comes to the door of his ready room.
Liza Minnelli is seen saying "Come on baby, Momma's gotta sparkle..." to an M&M. This is a reference to her drug and alcohol addiction that she went into rehab for in 1984.
Before the dog show, Brian acts like the director, Joe Gideon, in All That Jazz, using eye drops, taking a deep breath, and then saying "Showtime!"
After Peter tells Brian to "stop being a bad dog", Brian storms out of the house. Peter says, "Don't worry, he won't go far without this," holding an engine part, in a parody of the scene in The Sound of Music when the nuns reveal they have sabotaged the Nazi soldier's cars.
When Brian is kicked out of the Sicilian restaurant, he comes across the dogs from the Lady and the Tramp, in a parody of the spaghetti scene.
The family references The Match Game when they ask "How hot is it?" in response to Meg's comment about the heat.
Peter writes MacGyver asking him to save his dog; he sends him a drinking straw, a rubber band and a paper clip, referencing the character's ability to concoct ingenious uses for mundane household items.
During Brian's parole hearing, he references the court case Plessy v. Ferguson, a notable U.S. Supreme Court case that approved segregation in 1896.
The final scene, in which Brian drinks from the "people" water fountain, is a reference to the ending of the novel and TV movie, The Autobiography of Miss Jane Pittman, in which the title character, a black woman in the segregated South, does the same at a "white" water fountain.
The female dog showing her breasts in jail is a reference to the film Midnight Express.