Yesterday's Enterprise - S3-E15
Revealing mistake: When Tasha and Castillo walk up to the bar in Ten Forward, Whoopi Goldberg can be seen standing outside the door waiting for her cue to walk in the room.
Other mistake: One of the physical appearances Lal considers adopting is an Andorian female with green skin. In the Trek universe, Andorians invariably have blue skin (the Aenar are a variant of the Andorian race but are depicted with bluish-white skin, still not green). And while other humanoid races have been portrayed with different skin tones, there are some races that have only one skin hue. (ie, Andorians are always blue, Orions are always green, Ferengi are always light orange.) It is highly unlikely she or Data designed it with different skin as Data wants her to fit in, and Troi makes no mention of it.
Visible crew/equipment: While Troi tells Riker about the Captain's grueling negotiations between the Gemarians and Dachlyds, when Picard joins them on the bridge, there are three visible T-marks taped to the floor. (00:01:50)
Continuity mistake: While Tam is talking to Deanna and sits down, he says "You understood... at least a little, how I felt." While saying this, he places his left hand on his right leg. While still saying that same line, it cuts to an angle from beside Deanna and his left hand is suddenly between his legs.
Continuity mistake: To view their orders and mission, Data puts a drive into the Enterprise's computer while standing with Riker and Geordi. Riker is seen standing but leaning casually away from Data on Data's right, and Geordi is standing off to Data's left. It then cuts show the computer screen, and the three men are reflected in it. The reflection shows that Riker is suddenly almost leaning over Data's shoulder, and Geordi has moved to standing behind Data's left shoulder.
Revealing mistake: Just after Riker and Geordi are talking in the turbo lift, it cuts to a view from outside the ship and shows the USS Hood flying next to the USS Enterprise. As the ships draw nearer, the views of both ships overlap, and a black mat outline can be seen around the saucer section of the Hood as it passes by the Enterprise's starboard nacelle.
Continuity mistake: In the beginning of the episode, right after Data informs Picard that the USS Hood is on an intercept course, closing at high warp, Picard is shown in the back ground looking down at his arm rest. He looks up and turns his head to his right making it center with his body. It then cuts to a close up of Picard and he is looking to his left, and turns his head again to say "That's odd."
Continuity mistake: At the end of Barclay's sword fight on the holodeck, he has the hologram of captain Picard against a wall. When the angle is on the hologram, Barclay's sword is pressed into his chest. However, when the angle is on Barclay, the sword is several inches away from the hologram.
Factual error: The second piece of music played at the concert is by Brahms and not Mozart.
The Best of Both Worlds (1) - S3-E26
Revealing mistake: During the phaser battle on the Borg ship, a detonation wire is seen attached to the right foot of the drone that Beverly kills.
The Best of Both Worlds (1) - S3-E26
Other mistake: When O'Brien activates the transporter, only the very bottom indicator lights come on, even as he moves his hand up. Usually the indicator lights match the movement of the hand.
Answer: He brought the Borg to the Alpha Quadrant and showed them that it was full of worlds waiting to be assimilated. Guinan's homeworld was their first stop, and they assimilated everyone and took over the planet, leaving The Survivors of her race without a home. Q is ultimately responsible for that.
Captain Defenestrator
By the time Q takes the Enterprise to meet the Borg, Guinan already knew who they were and they had already destroyed her world. Therefore the above answer can not be right. I believe Guinan is much more than she appears, and her people have had encounters with the Q in the past. It is these interactions, that obviously were not pleasant, that fuels her distrust.
oldbaldyone
That's what the above answer is saying. Q brought the Borg to the Alpha Quadrant (not Earth) and the Borg destroyed Guinan's home world in the late 2200's, which is why she hates Q. Although she met Q in 2160 and they both saw each other as enemies right away.
Bishop73