Batman has Commissioner Gordon obtain 400 pounds of sodium dichloride; no such compound can exist, due to the chemical structures of sodium and chlorine.
Obvious dummies of Batman and Robin when activating their heal/toe bat rockets. The ropes lifting the dummies are visible.
The Riddlergram Batman received after the statue was destroyed ends with the riddle "What is it that travels on all fours in the morning on two legs at noon, and three at twilight?" As a classics scholar, Batman should have immediately recognized the Riddle of the Sphinx, famously asked of travelers by the Theban Sphinx and correctly answered by Oedipus.
Batman states that most of the survivors of the Titanic did so by treading water and staying calm. In fact, very few people were pulled from the water at all, as most of the survivors were already in lifeboats. Most everyone who went into the water froze to death. Even the information of the time would have been known to Batman.