Irreconcilable backstory inconsistencies, ship's scale and speed changing without explanation, set the show in the prime timeline but have a random background character dressed in a Kelvin-style uniform.
Sometimes phasers shoot beams, sometimes they shoot pulses. Parallel Earths everywhere. Flames shooting out the back of the nacelles for one episode then never seen again.
Do a time travel episode where they go back to the 1990s and the Eugenics Wars are definitely happening, then do another episode in the same era a few seasons later there they definitely aren't.
Have one of the characters be a huge fan of the TV show MacGyver, then do an episode where MacGyver appears and is treated as having been a real historical figure all along, then goes back to being a TV character again after that episode.
Establish a very specific and obvious naming convention for the ship's shuttlecraft fleet, maintain it rigidly for six seasons, then flagrantly disregard it without explanation come season seven.
Change the rules of the holodeck every time it appears in an episode. Sometimes holodeck food is real, sometimes it's holographic. Sometimes you can't walk into the walls while a program is running, sometimes you can. Etc.
Set the show in the early 25th century, but then have someone refer to an event that took place in the 21st century by saying "That was over five hundred years ago!"
One episode hinges on the ship getting lost in the epsilon quadrant, which is treated as though it's always been there, and then is never mentioned again.
Similarly, for one episode the captain of the ship appears as an animated character. This has nothing to do with the plot of the episode and is never acknowledged by any of the other characters.
Archive photos of Spock are seen at various times throughout the series, and each time Spock is seen, he's played by a different actor: Nimoy, Quinto, Peck, animated Spock, Jeffery Quinn, etc.
One episode makes reference to "Admiral Janeway's wedding." A wedding photo is briefly seen, where the bride is Kathryn Janeway and the groom is Tuvix.
One character is a member of an alien species with notably unusual feet, and the others are always talking about how weird his feet are, and then we finally see the feet and they're just normal human-looking feet and no one else even says anything about them.