"The Duke of Vermeer Caper" is all about the career of Johannes Vermeer, and how his incredible attention to light and shadow and surprisingly small body of work (only 34 known paintings) make his work highly desirable.
"The Opera in The Outback Caper" is set during a performance of Carmen (fittingly enough). The similarities of the characters is referenced often.
The car chase on the streets from San Francisco (from "The Lucky Cat Caper") is very reminiscent of the iconic car chase from Bullitt.
Professor Maelstrom's outfit (tan slacks, dark turtleneck, black jacket with white piping) is an exact replica of the one Patrick McGoohan wore in The Prisoner.
In "The Daīsho Caper", Boston natives Zach and Ivy sing "More than a Feeling" at karaoke, a song by the rock band Boston.
In "The Fashionista Caper", Agent Argent is practicing what she'll say to Carmen when she eventually catches her. At one point, she attempts to do a Horatio Caine-esque one-liner, complete with putting her glasses on and a song similar to "Won't Get Fooled Again" playing.
As of "The Need For Speed Caper", we can add Carmen Sandiego to the list of franchises that have paid homage to the famous AKIRA bike slide.
For that matter, the name of that episode is a shout-out to Need for Speed.
Also the episode where Paper Star attempts to steal the Magna Carta is "The Chasing Paper Caper", a shout-out to The Paper Chase.
As for Carmen Sandiego herself, the ability for her coat to transform into a hang glider is reminiscent to that of Kaito Kuroba. Bonus points that they are both thieves who use an alias to steal valuable items in order from them getting in the wrong hands.
According to "The French Connection Caper" A.C.M.E. agents drive Blanton-Webster automobiles, which in addition to being a Bland-Name Product version of Aston Martin, is also a reference to The Blanton-Webster Band.