Across whole show
Continuity: When the Villains send their cronies (for example, Putties, Tengu Warriors, Cogs, Piranhatrons, etc.), their numbers constantly change from when they arrive, to when they leave.
Visible crew/equipment: This happens mostly in the first 4 series - crew, lights, and equipment can be seen reflected in the Rangers helmets and visors.
Revealing: Until the end of the 5th series, we can see part of the Zord costumes move about as the character in them walks about - sometimes they even look like they are going to fall off. This is particularly true in the 2nd series.
Factual error: This happens until the end of series 3. Lord Zedd and Rita send down monsters to Angel Grove, in California. Yet when they cast their projectiles down to Earth (for example, with Zedd, a lightning bolt) they are seldom ever aimed even in the American continent.
Continuity: When the Rangers morph, the clothes they are wearing don't match what they have been wearing that episode - this also applies to hair.
Revealing: When the Rangers, and Villains fire projectiles at each other (for example, energy blasts) the ensuing blast suggests they were hit. However, when played in slow motion, we can see that the explosions shoot up from the ground around them - they don't even touch the characters.
All of series 1 (series 1)
Revealing: When one of Rita's monsters tears a building apart, the buildings are completely hollow, and nothing more than 6 pieces of Styrofoam stuck together to look like a building.
Continuity: Goldar's voice changes as the series progresses.
Plot hole: Up until about Green With Evil, we can see the Rangers fighting Super Putties, which didn't come into the Series until after Green With Evil.
Continuity: The clips on the side of the Rangers helmets constantly disappear and reappear.