The shades worn by Kevin Spacey were actually borrowed from U2's Bono who insisted that they be returned after the movie wrapped. See more...
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Continuity: When Prot takes his sunglasses off for his "return" (because he would supposedly no longer need them due to the lower light levels on K-Pax), he places them on the nightstand upside-down, with the earpiece flush to the table. Two subsequent shots (including an elaborate effects shot) show the sunglasses right-side-up.
Other: When we first see Jeff Bridge's character on the elevated subway, the car is jostling around in obvious motion. However, the scenary visible through the rear door window of the train is clarly not moving in response to the train. The city background is not moving farther away. The scene quickly changes angle and the background now begins to "move" in relation to the progression of the train car.
Plot hole: When the "Blue Bird" patient finally sees the Blue Jay, he starts excitedly shouting "Blue Bird! Blue Bird!" This uprising causes a riot in the institution, and in the immediate exterior shot of the building, everyone from all four floors already know and are staring out the window at the bird. Within the time-frame of the scene only the people on the floor which saw the bird would actually know about this. Psych ward floors are kept separated as it is even noted by an attendant when he said, "Do you want to go to the fourth floor?" which indicates that the floors are kept separate. This was deliberately done for dramatic purposes.