The Way of the Warrior (1) - S4-E1
Visible crew/equipment: At the start, after the Changeling surprise drill's staff is dismissed, while Sisko and Nerys are walking away, when she asks, "How about some dinner?" at the bottom left side of the screen, there are multiple T-marks visible on the floor behind her, ending with a pink T-mark as Sisko exits. (00:02:30)
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.
The Way of the Warrior (2) - S4-E2
Revealing mistake: During the fight in Ops, the laws of physics takes a slight dive. While Worf is fighting the Klingons, he hits them with his Bat'leth or his mek'leth, and a second later they react to the hit.
Factual error: When Major Kira was introducing Lieutenant Commanders Dax and Worf to the arriving Trill scientists, she introduces them as Commander Dax and Worf. While in informal conversation, they could be addressed as Commander, when making formal introductions, the person making the introductions would use their full proper rank. This is an error that no ranking officer would make, especially not twice. (00:03:10)
Suggested correction: And when was this established? If you are basing this error on modern American military protocols, there is no reason Starfleet in the 24th century would follow the same rules and traditions.
Yes, although it's not just American military introductions, it's pretty standard in most militaries to use full rank when making formal introductions, which has been followed for the most part by the writers, with other introductions that have happened onscreen.
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.
Factual error: The United States conducted no nuclear tests in 1947 and conducted none in Nevada until 1951.
Other mistake: Shortly after Quark, Rom and Nog arrive on Earth in the 20th century, just after they are observed through the one-way glass by the Army officers, the Captain in the hat who enters the room with the man in the suit to interrogate the Ferengi is wearing his rank bars on his shirt collar horizontally, parallel to his collar. The two bars of an Army Captain's rank should be worn vertically.
Visible crew/equipment: When Jadzia walks into Kor's quarters and finds Kor on the floor, just as she helps Kor stand up, there's a visible T-mark taped to the floor beside the chair. (00:08:35)
Factual error: As is typical for Star Trek maps of the world, New Zealand is missing from Dr. Noah's map of the world before anything is flooded. The Australian state of Tasmania is missing as well. (00:40:45)
Revealing mistake: Lt. Commander Eddington is wearing the rank insignia of a junior grade lieutenant.
Continuity mistake: It's surprising that Joseph Sisko is still alive at this point in the series. In previous episodes, Ben seemed to imply his father was deceased. In "Emissary" and "Paradise", he said his father WAS a gourmet chef (which is still his current occupation in this episode). And in a conversation with Odo in "The Alternate", Ben told him of how his father had a severe illness that made him very weak and he was unable to help him in the end.
Revealing mistake: When the Lakota opens fire on the Defiant, steam begins pouring into the Defiant bridge. The device releasing the steam is visible.
Other mistake: As Kira is showing Ziyal the Federation rifle, the compartment underneath the rifle flips open unintentionally.
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.
Continuity mistake: At the beginning of the episode when arriving at the station, Worf's brother has normal human bottom teeth. Later in the episode, he is wearing Klingon bottom teeth cosmetics.
Continuity mistake: In the scene where Dax is being questioned by the Klingon lawyer, Captain Sisko is shown having only 3 pips, or the rank of commander. (00:12:55 - 00:13:25)
Continuity mistake: During Worf's questioning, Worf switched from being a Lt. Commander (2.5) pips and a commander (3) pips several times.
Revealing mistake: As Sisko and Omet'iklan are arguing in the Mess Hall, the dead Jem'Hadar solider can be seen breathing. His chest is seen moving up and down. (00:30:35)
Body Parts - S4-E25
Visible crew/equipment: During a pan-across of Quark's bar after it has been emptied out, the shadow of the camera is visible for a moment in the bottom right corner of the screen. (00:40:35)