When Batman is the Batcave showing Robin the bug on Robin's glove, Robin's fist in clenched in the close-ups, but his fingers are extended in the wide shots.
After a henchman plugs in the Penguin-magnet on the penthouse balcony to trap the dynamic duo against the inner wall, the Penguin escapes via the same balcony. Behind him you can see the outer penthouse wall and the electric socket the magnet was plugged into, but the electromagnet is gone.
When Aunt Harriet comments on having seen Alfred and Dawn Robbins unconscious together, her tone implies Alfred was somehow responsible for the situation. However, Alfred was rendered unconscious by Penguin's gas immediately on answering the door, and thus was just as much a victim as Dawn Robbins.
When Batman and Robin are climbing the wall, it is obvious that Robin's cape is attached to Batman, and Batman's cape is being held up by a wire.
Penguin's giant magnet traps Batman and Robin by attracting their metal utility belts, but doesn't seem to have any effect on the metal in Penguin's umbrella and cigarette holder.
In the final moments, Dick Grayson, Bruce Wayne and Dawn Robbin's manager discuss the young woman's love for Batman. Behind Dick Grayson, a boom mic shadow can be seen moving along a display case and wall.