When Zoidberg takes off his helmet to eat the food in the first planet express ship, he should have died due to the rip in the hull seen in the background. When Zoidberg finishes eating, the hull is repaired.
When Professor Farnsworth is cutting Zoidberg's hair, the razor moves Zoidberg's scalp shell as if it was skin. Zoidberg is hard plated at the time and had his skull almost crushed in "The Silence of the Clamps" and "The Tip of the Zoidberg". Zoidberg could only have his skin move by an object if he molted but since he had hair on, it was impossible for him to molt with it intact.
Leela's obsession with killing a whale completely contradicts her pro animal protection movements.
When they pick up the statue, when not zoomed in on the text all the "N's" can be seen as "K"