In the opening scene, Bruce Wayne and Dick Grayson are in front of a blackboard with the Greek alphabet on it. As Dick is complaining about having to learn Greek, Bruce points to the following letters: beta, theta, pi. Adam West was a member of Beta Theta Pi fraternity at St. Louis University and this was obviously a 'secret' shout out to that by actor Adam West.
Ma Parker comes out of a theater which is showing "The Woman in Red," a reference to the killing of John Dillinger.
In one scene, Mad Dog hits Legs in the face with a grapefruit, paying homage to a scene from the 1931 movie The Public Enemy, where James Cagney hits Mae Clarke with a grapefruit.
Batman and Robin duck behind a drugstore counter. Behind them on the shelf are several boxes of the Toni perm kit. The boxes are all upside down, perhaps to avoid product placement.