Parturition - S2-E7
Revealing mistake: On the bridge, as Tuvok monitors and reports on the pursuing ship's weapons, you can see what looks like a wristwatch on his right wrist as he works the controls. (00:35:35)
Dreadnought - S2-E17
Continuity mistake: Dreadnought states there are fifteen priority targets approaching. On radar sixteen Rakosan ships are displayed. Then just before the Dreadnought attacks (and destroys three of) the approaching ships, there are a total of nineteen Rakosan ships displayed on radar.
Repression - S7-E4
Other mistake: In the 3D cinema scenes in the Holodeck, Tom Paris and B'Elanna Torres can be seen using blue/cyan anaglyph glasses and the movie screen can be seen to be in blue/cyan also. This is fine and to be expected. However the shot in which the projection room can be seen behind the couple show the projection is through magenta/green filters. (00:03:30)
Other mistake: They show Kes as a baby but her ears are human. (00:42:25)
Basics (2) - S3-E1
Deliberate mistake: After Voyager's secondary phaser banks overload, everyone on the ship seems to be severely injured, but Seska's baby is perfectly fine. This is likely to be a conscious decision by the writers. (00:40:30)
Mortal Coil - S4-E12
Continuity mistake: The shuttle is "parked" in the nebula to get a protomatter sample, and the nebula is visible in all windows. After the shuttle is hit, the camera looks at Tom, and the view in the window is the black of space with stars, rather than the bright nebula.
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.
Continuity mistake: In season 6 episode 10 'Pathfinder', Admiral Owen Paris has the rank of Full Admiral. In season 7 episode 6 'Inside Man', the same Character is seen as being only a Vice Admiral. In the season 7 season episode 'Endgame', the last episode of the show, he is once again seen as being a Full Admiral.
Revealing mistake: If you look closely at the hot air balloon, in the shot before the message is attached to it, you can just see a wire holding it up.
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