Several times during the series Harry Kim is given command of the ship during "night" hours. Yet Kim is only an Ensign in Engineering/Operations as denoted by his gold uniform. Command should have gone to Tom Paris as a Lieutenant in the Command (Red) function.
Seven Of Nine is frequently trying to become an individual. For example Star Trek: Voyager: One(#4.25), and in Star Trek: Voyager: Drone(#5.2), she is smiling at the beginning of the episode and near the end she cries (in fact it's the only episode where she does either one). Yet, except Star Trek: Voyager: Unimatrix Zero (#6.26), Star Trek: Voyager: Unimatrix Zero: Part II (#7.1) she continues to call herself Seven Of Nine instead of her human name, Annika Hansen.
Throughout the series, the number of crewmen on Voyager has fluctuated despite the fact that several have died over the course of the series, and only around eight have been added. The number of the crew has been as few as 125 and as many as 160. In Star Trek: Voyager: The 37's(#2.1 August 29, 1995) it's stated to be 125. See "Personnel changes" under trivia for specifics.
It's stated that Voyager has a "stable cruising velocity" of warp 9.975 but numerous times throughout the series the ship is shown "top out" at approximately warp 9.5 and is unable to go faster without damaging the ship.
Harry Kim is referred to as a "Senior Officer" in many episodes. Yet there are multiple Lieutenants on board and Kim is only an Ensign.
There is speculation that the way the Ocampa are shown to have offspring is an impossible situation, as a species where the female can only have offspring at one event in her life would half in population every generation, even if every single member had offspring. While Ocampa females can only become pregnant once in their lifetime, if was never stated how many children could be born at one time. Kes mentions having an uncle, implying that multiple births from one pregnancy are possible.
The Doctor is never given a rank. Since Data, an "Artificial life form" carries a rank, once the Captain declared the Doctor to be part of the crew, she should have assigned him a rank. However, the Doctor is unique and quite different from Data (who attended Starfleet Academy and earned his rank from years of working in Starfleet). Additionally, the Captain considered Neelix, Kes, and Seven of Nine to be part of the crew but they weren't given a rank either, so the commissioning of officers is by no means an automatic thing just because somebody is part of the crew.
Once the Doctor gets his mobile emitter, he is free to go about the entire ship and on away missions. When he is transported, his entire body shows the transporter effect like all the other characters.
But since the only part of him that is made of real matter is the emitter, only it should be shown transporting. The Doctor's image should simply disappear as the emitter is transported.
The doctor has several close calls with "death". For example: