The City on the Edge of Forever - S1-E29
Continuity mistake: At the end, Kirk says "Let's get the hell out of here." Captain Kirk and 6 others were beamed up at the same time, but the Enterprise transporter only has 6 pads; presumably a red-uniformed crew member was lost in space.
Suggested correction: Short version is, the Enterprise has more than one transporter room. The Animated Series references at least four on the Enterprise from this era. There are also cargo transporters, which aren't normally calibrated for personnel, but can be, albeit at reduced capacity and increased energy usage. https://memory-alpha.fandom.com/wiki/Transporter_room.
Is There in Truth No Beauty? - S3-E5
Other mistake: In the final scene, Kirk is still in the transporter room, but does not have on a visor when the ambassador transports out. Spock has his visor on, but then you see Kirk walk out of the transporter room without a visor and he is not going mad.
Visible crew/equipment: After McCoy examines Charlie and they talk about how Charlie learned to talk by just listening to the ship's tapes, when Charlie stands up the two actors' marks can be seen on the floor - the short one for Charlie and the longer one for McCoy, where they both will stand momentarily. (00:04:56)
Super Grover
Continuity mistake: When "good" Kirk beams up from the planet, the insignia on his shirt is not there. Then it appears when he's walking with Scotty. (00:03:00 - 00:05:25)
Garlonuss Suggested correction: This wasn't a continuous shot though, we just don't know where they get the insignia, it could have been in his pocket. Both Kirk and Sulu are on the planet's surface without their insignia. And when "evil" Kirk beams up, he too doesn't have it on, even though later he does.
Bishop73Deliberate mistake: Special effects of the ship in space were very expensive in the 60s and couldn't be wasted. So when a larger model was built with slightly different nacelles, shots of both versions became common, even within the same episode. This is why the Enterprise sometimes had red needle-tipped nacelles and sometimes lighted "spinning" ones, and in aft views she had either round white spheres or perforated vents at the nacelles' ends.
Jean GPlot hole: Pike says he doesn't want women on his bridge, Number One being the one and only exception, and he keeps ordering Colt off it. His apology to Number One indicates that A) he doesn't think of her as female, and B) she's the only exception to his rule. So how does he overlook the very female crew-woman seated at the science station? (00:06:55)
Jean GShore Leave - S1-E16
Visible crew/equipment: You can see the shadow of a boom mike on a tree just after the Caretaker appears. (00:44:05)
Continuity mistake: When Capt. Kirk and Charlie enter the turbolift heading for the bridge, Kirk is wearing one style of tunic and when the turbolift arrives on the bridge, Kirk is wearing a completely different tunic. (00:15:14)
Visible crew/equipment: When the doctor is mixing the drink and Pike gets up off the bed to accept it, an equipment shadow sweeps across the left side of the screen. (00:03:55)
Jean GVisible crew/equipment: After Kirk hits Kras with an arrow Spock's arrow hits a Capellan on the cliff, and when the Capellan falls into the bush, a crew member wearing sunglasses and a white T-shirt is visible at the left side of the screen. (00:43:50)
Super Grover 
Revealing mistake: A straight and level seam connecting the top and bottom halves of the tunnel Spock is crawling through is visible.
Scott215Tomorrow is Yesterday - S1-E20
Factual error: Towards the end of the show the Enterprise is leaving Earth orbit and heading towards the sun. We see the Earth diminish and the moon appear looking exactly as it does from Earth. From this angle we should be seeing the "dark side" of the moon, which looks completely different. (00:40:50)
vonThe Man Trap - S1-E2
Revealing mistake: Just after Kirk stuns Crater with his phaser, he gets up to get to Crater, when he does he bumps into a heavy stone monument,a prop that shakes visibly on contact. (00:36:58)
jbrbbtThe Gamesters of Triskelion - S2-E16
Visible crew/equipment: When Chekov's drill thrall sits down and nuzzles up to him in his cell, the boom shadow is visible on the wall at the upper left corner of the screen. (00:14:50)
Jean GCharacter mistake: Lt. Joey said to Lt. Sulu "You don't outrank me and you don't have pointed ears" but Joey is a junior grade lieutenant so Sulu would outrank him.
Where No Man Has Gone Before - S1-E4
Continuity mistake: When Kirk is fighting Gary Mitchell, his tombstone reads James R. Kirk. Kirk's middle name is Tiberius. Mitchell is a friend of long standing and would not have got it wrong. (00:44:30)
Richard WeltyRevealing mistake: When the laser cannon starts firing at the Talosian rockface, a styrofoam block falls and rolls away on the left side of the screen. After a brief shot of the crew, we see the cannon still firing, and the exact same block falls and rolls off screen on the left again. (00:30:15)
Jean GRevealing mistake: While the interior sets for the shuttle craft provided adequate room for the actors to stand, the mock-up built for exterior shots was undersized, and too short to stand up in by several inches. So when entering or exiting the shuttle, the actors always had to duck.
Jean G
Mudd's Women - S1-E7
Continuity mistake: As the women are beaming aboard in the transporter room, McCoy's uniform switches from the standard blue tunic to his short-sleeved medical smock and back again between shots. (00:05:55)
Jean GRevealing mistake: As the bridge crew monitors Psi 2000 on the view screen, a glitch in the special effects matte makes the planet vanish briefly and then reappear - twice. (00:12:30)
Jean G
