Plot hole: To escape the 'event horizon', a rupture in it is widened to 120 meters (which allows 2 meters clearance on each side of the ship). Soon after, the rupture has shrunk to 110 meters wide, and Janeway decides to 'punch through' anyway. The ship does so with little difficulty, but the whole issue seems pointless since the rupture was far larger 'vertically' than 'horizontally' (relative to the view from the ship), so tipping the ship on its side would have allowed plenty of clearance to slip right through.
Revealing mistake: If you look carefully when Neelix shoots the bins of water causing the water to rush out you can see a faucet opening on one of the bins.
Continuity mistake: The registry on the shuttle that brings Paris to Voyager keeps changing between "NCC 71225" and "NCC 1701-D".
Continuity mistake: Tuvok and Paris' rank seems to skip around during the first season. Despite being listed in the credits as 'Lieutenant Tuvok', Tuvok starts the show as a Lieutenant Commander, with the signatory three collar pips, one black and two gold. At certain points, however, he is seen with only two gold pips (signifying Senior Grade Lieutenant). After season one he remains a Lt. Commander with no reason for the lack of continuity offered. Similarly, Paris starts the show as a Senior Grade Lieutenant, but sometimes instead of two gold pips he has one gold and one black, the signifier of a Junior Grade Lieutenant. After season one he is always presented as a Junior Grade, excluding his brief demotion to the rank of Ensign in a later series.
Emanations - S1-E9
Factual error: To protect the warp core, Janeway instructs Paris to move away from the rings at Warp 7. Moments later (Janeway, Torres and Tuvok are still in the same position), Paris tells Janeway they are 0.6 light years away from the rings. Warp 7 is 656 times the speed of light. It would take about 8 hours to travel 0.6 light years at Warp 7.
Suggested correction: The warp scale, especially since it's fictional, has never been precisely defined.
While it's never been defined, certain aspects of warp speed drive and the length of a light year are accepted. Even if Warp 7 is double what is suggested in the mistake, it would still be 4 hours. The scene in question shows no lengthy travel at Warp 7.
Visible crew/equipment: Neelix says to Chakotay 'I don't understand, it's right here,' referring to the dilithium ore. As soon as the camera changes to show him walk into the large cavern, a boom microphone is quickly pulled away from covering his face. (00:07:45)
Continuity mistake: In sickbay, when Kim tells the captain he's analyzed the tricorder, the tricorder is open. In the next shot it's closed.
Revealing mistake: Tuvok is behind a forcefield in Sick Bay, with the Doctor, Kes, and Captain Janeway looking at him. He tries to break the forcefield and is rendered unconscious and thrown to the floor. Janeway and the Doctor rush to pick him up and put him on the bed. Before they start to pick him up, he lifts his head and starts sitting up (to assist them, not because his character is conscious). (00:38:26)
Factual error: The event horizon of a quantum singularity (aka black hole) is not a physical barrier you can punch a hole through but a theoretical one, which can't be escaped from without faster then light travel.
Character mistake: When trying to trap the energy mass they created, B'Elanna announces, "Now it's trapped on three sides," to which Paris replies, "It's changing course. Heading for the open side." Since we function in 3-dimensional space, unless they are trying to trap it in a triangular-based pyramid, the number of "sides" to be concerned with is six.
