Deliberate mistake: The dying Vantika hides his consciousness away in a portion of Dr. Bashir's mind. At one point, "Bashir" (heavily cloaked in clothing and shadows) comes up behind Quark and makes sure his demands are clear. At this point, it should still be unclear just how Vantika is still "alive", and whose body he is using. However, when he shoves Quark away, for a fraction of a second, Bashir's face can be seen under his hood. They were true to plot in using the right actor (though he shouldn't have been seen), it is possible some might have spied the secret earlier than the show meant to reveal it.
Revealing mistake: As Kurn is chastising Worf in the infirmary, the actor's prosthetic nose has become detached from the rest of the Klingon make up. You can see the nose wobbling around as he talks.
What You Leave Behind (2) - S7-E26
Quark: The more things change, the more they stay the same.
If Wishes were Horses - S1-E16
Trivia: Buck Bokai's baseball card, a collectible featured on Benjamin Sisko's desk, had actor Keone Young on the front, in character, but showed "Trek" model maker Gregory Jein, who invented the "history" of the character, on the back. The pair bore an uncanny resemblance to each other.
Emissary (1) - S1-E1
Question: Did Nana Visitor's hairstyle change from her first appearance in the episode to the next scene when they were going down the stairs?