Encounter at Farpoint (1) - S1-E1
Revealing mistake: When Q freezes Torres on the bridge, it is easy to tell that directional hoses were used for the effect. (00:06:25)
Adventures in Turtle-Sitting - S6-E4
Revealing mistake: When Leonardo throws his katana at the tree which Michelangelo and April are tied around, you can see the rope is cut and begins falling off the tree before the katana touches it. (00:20:33)