Non Sequitur Scene: If you actually try to focus on the plots of the shows that seem to have one, then almost every performance is one big-lipped alligator moment after another.
Come for the X, Stay for the Y - The surreally-staged acrobatics are what the ads emphasize, and rightly so, but the music, comedy, and characterizations are often just as wonderful, providing much of the shows' repeat value.
Follow the Leader - Imitations of this company's style exist, such as the output of the U.S.-based Cirque Productions (unsuccessfully sued for using "Cirque" in its name). Can qualify as The Mockbuster when one considers that these shows usually travel to places that don't usually, if ever, get actual Cirque tours...
Genre Turning Point - The company for circus in general, Mystere (in tandem with Siegfried and Roy's show at the Mirage) for Las Vegas entertainment.
Germans Love David Hasselhoff - Cirque is popular enough in Japan that the Kaleido Stage is clearly inspired by it, and with ZED, it became one of only two countries outside of the U.S. to get a resident show (its closure was only because the earthquake/tsunami of 2011 caused a decline in tourism and thus audiences). This dates back to the 1992 Japan-only tour Fascination.
Magnum Opus - "O" for the company as a whole; Alegria is a worthy runner-up and holds this position among the tours.
Ruined FOREVER - As a columnist for the online fan newsletter "Fascination" pointed out, the company has been getting this complaint from fans as far back as Nouvelle Experience and its more theatrical approach, though complaints really started picking up in 2008, as more and more new shows were announced/mounted and ZAIA and Criss Angel Believe opened to underwhelming response. The announcement of the Michael Jackson shows in 2010 was received even more poorly, partially because Jackson hadn't been dead a year and it thus looked like a cash-grab.
Which is funny since the Michael Jackson shows have been getting the best reviews for their arena tours since Alegria. This really is more of a case of Hate Dumb more than anything.
True Art Is Incomprehensible - It's possible to enjoy these shows without being aware of their underlying themes (individual interpretation is encouraged), but it isn't always easy.