The film is supposedly set in 1932 because of Scout's narration stating she first knew her hometown in 1932. If this is indeed the correct year, the PF Flyers Jem and Scout are wearing would not have been available for a few more years. I would suggest, however, that the setting is in fact a few years later after the FDR quote about "fear itself" was delivered in 1933.
When Scout and her brother are going to the pageant, she's dressed as a ham. In the close-ups you can see that the "ham" is actually made of fiberglass. Fiberglass was invented in 1938 while the film takes place in 1932. However, it could be made of painted canvas or burlap on a chicken wire frame; in any case, it is light enough that Scout carries it with one hand at 1:51:29 as she and Jem approach the school auditorium. See close-ups at 1:53:51 as Jem shushes Scout to listen and at 1:56:10 of just the viewport.
When Jem and Scout leave the tree with the knothole in it when Mr. Radley is filling in the knothole, he leaves his school books at the foot of the tree. At 0:53:59, Jem drops his books by the tree to adjust Scout's feet to "walk like an Egyptian". At 0:55:02 they run off without the books.
In the very beginning of the film, the adult Scout's narrates, "Maycomb was a tired, old town, even in 1932 when I first knew it." Moments later, she goes on to say that "Maycomb County had recently been told that it had nothing to fear but fear itself" (at 0:03:47) in clear reference to Franklin D. Roosevelt's first inaugural address. Roosevelt, however, did not deliver this speech until March 4, 1933.
In the final scene, the narrator says "He (Atticus) would be in Jem's room all night and he would be there when Jem waked up in the morning." Atticus may have stayed all night in the room where Jem was sleeping, but it wasn't Jem's bedroom; it was probably Atticus' bedroom considering all of the law books about the room.
The license plate on Atticus Finch's car changes during the movie. At first it is 709-865 (at 0:44:20) as Atticus backs his car out of the garage to drive to the Robinsons the first time. Later in the movie, it is 369-883 (at 1:47:57) as Atticus stops at the Robinsons a second time.
When Jem shows Scout all of the things he's collected in the cigar box (0:55:15 to 0:58:00) the curtains in his bedroom are different than the ones in a later scene when he wakes up to follow Atticus downtown (at 1:00:31). Between these two scenes the narrator says "School finally ended and summer came and so did Dill." Then two brief additional scenes separate them. The curtains are light and appear to have the same design but look fuller in the later summer scene where they are half closed than in the earlier scene where they are fully open, as though one wouldn't be able to push them as flat against the window frame in the later scene as in the earlier scene due to bunching on the curtain rod which was out of frame in the earlier scene.
When the rednecks from Old Sarum arrive at the jail alleyway to lynch Tom Robinson the rear shot of them approaching Atticus shows a very young lout in the group holding a rifle (at 1:02:19 in a light shirt with sleeves rolled to his biceps and no hat), but in all the following shots from the front he's nowhere to be found.
While Atticus gets his papers together in the courtroom after the verdict you see a water glass next to the pitcher on the judge's desk (at 1:42:43). In the next shot, as he walks out, there is no glass, just the pitcher (at 1:42:48 and 1:43:06). Similarly, at 1:10:58, the glass is setting slightly behind the pitcher as Sheriff Tate confirms Mayella was beaten about her right eye, but is not there at 1:11:07 as Atticus walks up to Sheriff Tate. Note that when the glass sets further back than the pitcher, it can be concealed by the pitcher when viewed from some angles.
Scout steps up with both feet to stand on the tire swing to look up into the tree (at 0:05:51 when Jem won't come down from the treehouse). The next shot shows Scout standing on the ground and stepping up into the tire swing again.
When Mr. Gilmer gets up to cross examine Tom Robinson, he walks past the seated Atticus at the defense table (at 1:29:52). You can only see the arm of the person as he walks by the table. The person who walks by is wearing a short-sleeved black t-shirt (he is probably a production stand in). When Mr. Gilmer reaches the witness, he is wearing the same light colored suit he has worn throughout the entire trial, which covers his entire arm.
During the courtroom scene, there is a close-up of Atticus (at 1:12:50) seated at his table during the prosecuting attorney's questioning of Mr. Ewell. In this shot, the film image is reversed (Atticus' hair and the position of the spectators behind him reveal this). In this same shot, the light at the back of the courtroom is turned on; both before and after this shot, it is off. Actually, the light that is on in the close-up shot (at 1:12:50) is not the same light that is off in the wider shots of Atticus and Tom seated at the defense table (such as at 1:12:07). In the close-up shot the light that is on is five to ten feet from the double door court room entrance with no windows visible in the shot. In the wider shots, the light that is off is five to ten feet from a window with no doors visible in the shot. The close up was flipped accidentally or intentionally either "because" or "so that" the resulting frames more closely resembles the frames of the wider shots.
When Bob Ewell is on the witness stand and Atticus asks him if he ran for a doctor, Atticus is hovering over Ewell and his visible shadow behind Ewell reflects this (1:13:52 to 1:13:56). When the camera shifts to Atticus, he is a good 10-15 feet away (1:13:56 to 1:14:00). When the camera returns to Ewell, Atticus' shadow is still there (1:14:00 to 1:14:10).
While Atticus is giving his closing arguments standing before the jury, the camera briefly switches to Jem (at 1:38:21) sitting in the balcony and then quickly back to Atticus (at 1:38:24) who is now leaning on the rail in front of jury instead of standing stock straight. The cuts are not seamless. At 1:38:20 a close-up of Atticus ends where he is only seen from his elbows up and his thumbs may be tucked into the pockets of his vest. After the close-up of Jem, the scene cuts to a long shot of Atticus from the back of the courtroom standing on his left leg with this right leg resting on the bottom rail of the jury box, perhaps leaning with a stiff right arm on the top rail of the jury box and with his left arm across his chest perhaps grasping his right arm - in this pose, he says "In the name of God, believe Tom Robertson" and pauses for three seconds, at which time (1:38:36) the scene cuts to a close-up of Atticus from above his elbows up, apparently leaning with both arms on the top rail of the jury box, raising himself to an upright position.
While lying unconscious in his bed after Boo Radley carries him back home, Jem's head is seen in several different positions. (After comparing the images from 1:58:39 to 2:02:05 this seems to be somewhat exaggerated. There are small changes, but his head is generally turned 45 degrees to his left.)
While Atticus is questioning Mr. Ewell in the courtroom (from 1:13:33 to 1:15:25), Mr. Ewell has his left arm draped over the back of the witness chair in some shots (at 1:13:59) and by his side in others (at 1:14:00).
When Atticus is sitting in front of the courthouse door to protect Tom Robinson, the bottles in the top crate in the corner change from fewer bottles before the mob arrives (four are visible at 1:01:44), to more when the mob has left (five are visible at 1:06:24). The top crate has also been rotated about 30 degrees, so the fifth bottle may have been hidden by one of the other four in the first shot.
In the long shot when we see Jem discover the spelling medal left in the old tree it's sitting in the lower right hand side of the knothole (at 0:50:11), but in the close-ups it's sitting in the middle of the opening (at 0:50:23).
When sitting on the porch with the Judge, Gregory Peck is shown sitting on the porch swing with his left arm on the arm-rest. However, during the solo shots of the judge it is obvious that no-one is sitting on the swing - no arm on the arm-rest and the swing's chains are loose (from 0:16:40 to 0:17:50).
When the children are shown waiting for Atticus to come home, Jem is lying in the tire of the rope swing counting each time the swing changes directions while we hear the courthouse clock chime seven times. As the clock is chiming for the sixth time, Jem slips out of the tire and starts running down the sidewalk yelling (at 0:11:17) "Come on Scout! It's five o'clock."
After Atticus speaks with Mrs. Dubose at her porch, Scout, Jem and Dill walk away with Atticus towards the Finch home (at 0:13:00) while the camera pauses on Mrs. Dubose pondering Atticus' words. When the camera cuts back to the Finches (at 0:13:05) they have walked over to their house, up their sidewalk, and are at the steps to their porch while Dill has gone past their sidewalk, past their driveway and is standing by the rope swing in their side yard looking back at the Finches as if he were watching them arrive home. It looks awkward, as if it were taken from another scene where they were arriving home, but it could have happened that way if Dill were too shy to accompany them up their sidewalk without being invited (which, given Dill's personality, doesn't seem very likely).
When Atticus carries the sleeping Scout down the hall to her bedroom, after returning from their visit to the Robinson's, he takes her all the way to the last room on the left, which is actually Jem's room. Actually, it is a hallway. Calpurnia is holding the door to the house open when Atticus reaches the porch revealing a door with a 15-pane French window at the end of the entryway (at 0:48:03). Atticus carries Scout all the way to this door and turns left, not into a bedroom but into a hallway to the children's rooms. Their bedrooms were shown when Scout looks at Atticus' watch: the camera is pointing from Scout's window, across Scout's bed at the solid door to the dining room in the center of the frame, just beyond Scout's table lamp (at 0:13:28), with the dining room visible in the background on the right side of the frame and the wall behind Scout's headboard along the left side of the frame. After kissing Scout goodnight and turning her lamp out, Atticus swings the dining room door from our left to our right (at 0:15:19) revealing the solid door to Jem's bedroom which he enters walking to our left, tells Jem goodnight, shuts off Jem's light, comes out, closes Jem's door and walks down a hall to our right, which would be the hall into which he turned at 0:48:10 carrying Scout.
The law books disappear from the mantelpiece when Scout returns from walking Boo Radley home (at 2:08:17). This would be the bedroom with a fireplace and a woman's photograph on its mantel. The law books were never on the mantel, they were on a dresser (or tall writing desk) beside the bed (at 1:58:26). In some shots, they seemed to be on a mantel (at 2:01:36) because the dresser is cropped off by the bottom of the frame.
At the end of the movie Scout reminds us of the gifts Boo Radley has given them over the years, including "a broken watch and chain." However, during the opening credits, the watch face looks intact and the watch is ticking loudly (at 0:00:52 where the watch has no hands -fake hands have been drawn on the face of the watch - hence the watch is broken). The watch is also visible and ticking softly when Jem first shows Scout his box of things he found in the knothole of the tree in front of the Radley house at 0:56:52.
When Bob Ewell testifies he says he saw Tom Robinson running out the front door (at 1:12:46).
When Mayella Ewell testifies she says (at 1:17:06) when she came to, her father was standing over her yelling who done it?
Both stories can't be true. But neither Atticus Finch nor anybody else points this out.
Similar to her "close up" conversation with Jem about retrieving his breeches, Scout mouths Atticus' next lines when begging to be allowed a ride to visit Tom Robinson's family before the court case (from 0:44:27 to 0:44:57). The lip movement is slight and would be mumbling were she speaking, but when Atticus says "Promise to stay in the car while I talk with Helen Robinson" the movement does seem to match "Helen Robinson".
Directly after the scene where Jem and Scout are attacked while walking home through the woods, as Scout runs after the figure carrying Jem home, the trees and scenery can be seen through Scout in a ghostly fashion as if they were not originally part of the scene and were added afterward. Director Robert Mulligan mentions in the DVD commentary that this is the only special effect in the movie (at 1:56:39). This was necessitated because the extended shot shows the transition of Boo and Scout from the woods to the Finch house, because everything was shot at Universal Studios (Universal City, California), and because there were no woods near the studio recreation of Maycomb.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
After the attack on Jem and Scout, Boo carries Jem towards the house and eventually is discovered standing behind the door in the room where Jem is lying. However, in the scene before he is discovered (1:57:50 to 1:58:11) the door to the room is almost closed completely and then opened by Calpurnia from inside the room. Boo would have to have been visible to all in the room at that time. It is wrong to assume that the doctor, Calpurnia, and Atticus (who were in the bedroom in the previous scene mentioned) didn't know who carried Jem to the house. It was when Scout was describing to Sheriff Tate what happened and she mentioned seeing someone carrying Jem that the sheriff asked her who it was and Scout pondered a bit, but then saw him standing "behind" the door and said: "Why, there he is Mr. Tate." Those who knew Arthur Radley may have known he was not capable of telling them anything about the incident.