Character mistake: In the scene where Lwaxana Troi follows Odo into the Turbo lift and it breaks down, Odo reports to Ops that they are in Turbo lift 7. Later scenes in Ops refer to their location as Turbo lift 4.
Second Sight - S2-E9
Character mistake: When Nedelle collapses, everyone looks to Jadzia to save her and she says she can't. Why do they think Jadzia can save her? She isn't a doctor, and the Dax symbiant was never a doctor either. They are on board a Federation ship, with a Federation crew, why do they not call The Ship's doctor?
The Way of the Warrior (1) - S4-E1
Character mistake: At this point in time, the Khitomer Conference (where the Federation and Klingons resolved their conflict) was at least 70 years ago, yet Bashir refers to their peace with the Klingons as having lasted two decades.
The Way of the Warrior (2) - S4-E2
Character mistake: During the Defiant's flight to save the Cardassian officials, Worf says something to the effect of, "I've never been aboard a Federation ship equipped with a cloak before". However, he was aboard the Enterprise during the Pegasus incident, where The Ship was temporarily equipped with a cloaking device.
Character mistake: Sisko states they are attempting to create the first artificial wormhole (as in constructed). The problem is, the pilot episode over three years earlier established that the Bajoran wormhole wasn't natural, but constructed by the Prophets.
Character mistake: At the end, Sisko blames the deaths of the runabout crew (Hoya, Rooney and Bertram) on himself and the Vorta not trusting each other. However, the runabout was destroyed before any negotiations had taken place, and even before any Jem'Hadar or Vorta had landed on the planet, so their deaths were largely unavoidable. Even the later deaths of T'Lor and Muniz were caused by the first firefight, again, before any negotiations had happened.
Suggested correction: Alternatively, Sisko could also be referring to the general mistrust that exists between Starfleet and the Dominion, which is constantly threatening to boil over into full scale war. Without that prejudicial mistrust and paranoia, The Jem'Hadar would have likely never killed Muniz or attacked the runabout. Indeed, it can be said both sides inability to negotiate throughout the episode is due to their mutual belief that the other side cannot be trusted under any circumstances.
Doctor Bashir, I Presume? - S5-E16
Character mistake: In the scene in Sisko's office where he has contacted Rear Admiral Bennett, Bennett says "Two hundred years ago, we tried to improve the species through DNA resequencing, and what did we get for our trouble? The Eugenics Wars." This puts the wars in the 22nd century, and differs from every other mention of them, where they occurred in the late 20th century, not the 22nd century as Bennett implies. (00:39:30 - 00:40:00)