Continuity mistake: When Tuvok and Paris go scouting the holodeck, Chaotica's fortress is visibly damaged, but in a later shot when Kim and Seven are looking at it from astrometrics, the building looks like it's completely intact.
Revealing mistake: Right at 40:30, if you look in the top left corner of the LCD screen, you can see a mouse cursor. Six seconds later it starts to move around. It seems like someone noticed and tried to move it out of the scene. (00:40:30)
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: After Starling is beamed out of the limo, Rayne opens the door and escapes. The door is still standing open when the chauffeur starts the car. But in the reverse angle as he drives away, the door has closed itself. (00:20:10)
Continuity mistake: When Tom explains to Alice why he can't leave the Voyager, he has a three-day beard. Later in the ship, when Alice convinces him to turn on the neurological interface, he is cleanly shaved. But in the next scene he has his beard back again.
Revealing mistake: When Tom is reading the Klingon romance novel, he presses the button to turn the page, yet the text is always the same.
Plot hole: Tuvok says that he was born on Stardate 38774. But he was born in 2264, when Stardates were only counted in the thousands, not the tens of thousands. If he were really born on SD 38774, that would mean he was born only a few years before the Enterprise-D was launched.
Visible crew/equipment: When the camera turns to face Tuvok as he approaches the crazed Hirogen, two stage crew members can be seen at the left side of the screen. The first is a rather plump man wearing what appears to be dark pants, and a short sleeve light shirt, while the second is squatting down in front of the first wearing shorts. (00:09:05)
Character mistake: Neelix says that the Great Wall of China, prior to the 22nd century, was the only man-made object visible to the naked eye from orbit. This statement has been long-since debunked, but it's not unreasonable to think the myth perpetuated itself in the future. Still wrong though.
Continuity mistake: Paris reports that several ships have dropped out of warp off of Voyager's port bow, but when the ships are put on screen, they are approaching from starboard aft.
Basics (2) - S3-E1
Plot hole: Paris asks the Doctor to disable the phaser banks. How can the Doctor, who can't exit from sick bay, disable banks in engineering? And if it is possible to disable them from sick bay, why did Suder go to engineering?
Continuity mistake: Early in the episode, the Kyrian historical researcher plays a simulation of events aboard Voyager that include Janeway killing a Kyrian and then throwing the rifle to a crewman. Later, he replays the simulation for the Doctor, but subtle details of this 'same simulation' change, most notably the way the female crewman raises her rifle much sooner the first time we see Janeway walk past her, and the difference in the way the male crewman catches Janeway's rifle. (00:14:15 - 00:26:20)
Other mistake: When Seven talks to Lt. Com. Tuvok when he is shaving, she refers to him as "Lieutenant," rather than "Commander." Starfleet follows American naval doctrine with respect to rank structure and military courtesy - Lieutenant is not an acceptable abbreviation of Lieutenant Commander, and with the Borg having assimilated Starfleet vessels and individuals in the past, she'd know their rank structures.
Other mistake: In this episode Voyager sends the doctor to a federation ship in the alpha quadrant through a transmission using an alien communication network. Right before he is sent you see him in the astrometrics lab and he steps up on a platform wearing his mobile emitter. His program is then sent through the data stream and the mobile emitter is sent with him. Then when he arrives it is not there. Firstly, the mobile emitter can't be sent this way, only the doctor's program can. It can be beamed but that was not the case here. Secondly, in other episodes where the doctor is taken offline while wearing the emitter they show it drop to the ground. They don't do that here as it appears to dematerialize with him. (00:07:00 - 00:08:00)
Revealing mistake: When Paris and Rayne are driving through the Los Angeles suburbs in her van, they never change their direction of travel, but they pass the same sequence of houses twice. (00:31:05)
Visible crew/equipment: After Paris hails Tuvok about the problem in the mess hall, a brawl breaks out. When the new Hirogen Alpha enters the galley to access the com relay, just as this Alpha removes the front panel in order to transmit a signal, the actor's green tape mark is visible on the floor. (00:39:00)
Visible crew/equipment: When the former Borg drones are awakened in sickbay, there's a pinkish T-mark visible on the floor behind Seven, at the bottom left corner of the screen. (00:19:10)
Other mistake: When Janeway plays the message from the Admiral, she tells the computer to advance to time index 121.4. each 0.1 of the time index equals 1 second. We hear the message play for 4-5 seconds, but when the video is shown, the time index is only 121.5.
Continuity mistake: When future Kes is talking to Neelix in the kitchen, when he says he's already reserved the holodeck, he's holding the smaller container. In the next shot, the smaller container is on the counter and he's holding the bigger pot.
Character mistake: When the now human Borg children are sculpting, one of the children tell Seven he made a cube that's precisely 1/1,000 the size of a Borg vessel. Borg Cubes are described as 3 kilometers in length, if not bigger. 1/1,000 of that would be more than 9.8 feet in length. 1/10,000 the size would have been a more accurate description.
Answer: If you include the original Star Trek series (1966) then there are several. The communicators used in the original series were before (and said to inspire) mobile phones. We currently do have teleportation technology but it currently only works on things the size of a few molecules. A "Cloaking device" also exists; it's a fabric that bends light through it, though it currently only works in infra-red. The Hypospray is real and was patented in 1960 - six years before the original series aired - it's actually called the Jet Injector. Faster Than Light travel is still a few decades off, but there are several real-world theories that look promising, including one that is remarkably similar to the method used in the Star Trek Universe called the Alcubeierre Drive that involves manipulating spacetime ahead and behind the ship and the ship "riding" it. Medical techniques and technologies have also advanced considerably; prosthetics particularity and we routinely have robots performing surgeries where absolute precision is needed. The "Shield" used in the series have a few primitive versions around. The Phasers used in the series are used but are not very powerful (nor will they ever be as powerful as the Star Trek version the laws of physics gets in the way) but rail-guns (using magnets to spin then propel a projectile) and particle accelerators like the Large Hadron Collider have been around for a while. The Replicator would require a nuclear fusion reactor and a nuclear fission reactor in something the size of a large oven and the Holo-deck wouldn't work at all based on our current understanding of physics so those are both still science fiction at the moment, but who knows!
Sanguis