Cole is aware of his teacher's stuttering issue during grade school through high school (Stuttering Stanley), but the behavior has nothing to do with anyone being dead, so it doesn't make sense that Cole knows this.
Malcolm is translating "De profundus clamo ad te domine" and begins "Out of the depths". But he's translating the phrase word by word from a Latin dictionary, and it's more natural to translate "De profundus" as "From the deep" before refining the phrase after the whole of it is understood.
When Cole is walking with Malcolm through the streets of Philly to go see the German man, the streets and sidewalks are covered with brown fall leaves, but the trees along the street are all green as if it is spring time.
When Malcolm and Cole meet in the church for the last time, a patch of hair is sticking up from the right-hand side of Cole's head. The hair becomes less frisky between shots until it ends up well combed.
The first time Malcolm meets with Cole, right before they leave, Malcolm's arm is on the back of the pew from Cole's point of view, but from Malcolm's point of view, it is down. Right after Cole grabs the statue of Jesus, he puts his arm up again.
During the traffic accident scene, Pennsylvania state vehicle inspection stickers (which are required in lower left corner of driver windshield) are visible on the cars in front of Cole's as the camera dollies towards his car. But his mother's Volvo has no stickers even though earlier in grocery store parking lot the same car did display PA plate.
When Malcolm is waiting to meet Cole for the first time, we get a shot of the paper he's been writing, and a good look at certain circled words that compare Cole's case to an earlier case. Later, after their meeting, we get a chance to read that paper, and it becomes obvious that the words he had already written were actually a description of the event yet to occur: his first meeting with Cole.
The use of "visible breath" to indicate the presence of a ghost or evil is inconsistent; especially when referring to Dr. Malcolm Crowe (Bruce Willis). Since he is dead, it should be cold whenever he is with anyone.
The goof items below may give away important plot points.
At Kyra's funeral, Cole enters Kyra's bedroom through a door to the left of the bed, but on the video of her mother poisoning her food, her mother enters the room through a door to the right of her bed.
At the end of the film Malcolm Crowe comes down the stairs to the living room. The camera is at a low angle looking up from the living room floor toward the vestibule area at the foot of the stairs. There is a leather covered sofa and two overstuffed leather chairs crammed side by side to fit into the space. Crowe's wife is slouched in one of the chairs. The camera changes to above, the wife falls asleep and drops Crowe's ring, which now rolls over open floor boards. Where did all the furniture go?