(at around 1h 4 mins) When Agatha enters the Nathaniel Institute, it is dark outside, but when we see her again in the doctor's office, bright light is shining through the windows. When she leaves again it is dark.
(at around 1h 10 mins) After Walter and his son have found the buried skeleton only to return with the police to find it gone, they are met by his wife, whose hands are clean. However as they drive home Walter notices her hands are very dirty.
(at around 1h 1 min) When Walter Sparrow enters in the post office while Dr. Sirius Leary is opening the boxes, we can clearly see sheets of Neutral Density filters placed on the windows behind Walter Sparrow.
(at around 41 mins) In the film, Walter Sparrow claims that the serial killer Ted Bundy died on January 23. Ted Bundy was actually executed on January 24, 1989. However, he was pronounced dead at 7:16 am (7+16=23).
In real numerology, numbers are added together to form a single digit number that might have significance. The sum should never exceed 9, except for the "master" numbers 11 and 22. This is in accordance to the nine personality types, the nine planets in the solar system, and many other things. For example, Walter's room at the asylum was 318. When calculated in the movie it is suppose to equal 23, 31-8=23. However in real numerology the number would be added as 3+1+8, equaling 12. The sum would then be added again to create a single digit, 1+2 = 3. 3 would be the significant number here, and there are many sources on the Internet or in various numerology books that would define what significance that number would hold for the individual. As another example, the address of the bookstore would have more credibility as their sum does equal 23 (5+9+9=23). But another step would need to be taken to convert the sum into a single digit (2+3 = 5). Finally, birthdays and other notable dates in numerology are always factored with the year. Following the formula of February 3 (2/3 = 23) would be inaccurate. But December 12, 1975 (12/12/1975 = 1+2+1+2+1+9+7+5 = 28 = 2+8 = 10 = 1+0 = 1. Therefore, for a person born on this date, the number 1 should be the most significant.
(at around 37 mins) A segment in the opening credits states that the Mayans believed the world would end on December 23, 2012, while in the film, Danny Huston's character reads that the Mayans believed that the world would end on December 12, 2012. In actuality, the Mayans did not believe the world would end in December 2012; they believed a new world would be created. (The date of this is still widely disputed as well, with some sources placing the date as December 21, 2012.)
The goof items below may give away important plot points.
At some point in the movie, it is revealed that the actual book written by Walter Sparrow was written in 1991. There is one scene where he is reading the book and reading off dates of evil occurrences that add up to 23, and one of those dates is the Oklahoma City bombing, which happened in 1995. Since he is reading from the book that he wrote in 1991, that bombing would not have occurred in time to be included in the book.
(at around 55 mins) The final chapter of the book reveals that Walter met Laura at college, and that she was having an affair with her professor - the same man who was wrongfully convicted of her murder. But when Walter visits this man in prison, there is no indication that he recognizes Walter, which he presumably would have.