The movie is set in 1931. From 1925 to 1934 the Eiffel tower had illuminated signs for Citroën adorned three of the tower's four sides. However in the movie the lights on the tower are as they are today, with no Citroën sign on it.
The movie is set in 1931. But the Django Reinhardt character is shown with a Selmer Maccaferri oval-hole guitar, which was not introduced until 1936. Also, he looks a bit older than the real Django, who would have been just barely 21.
In the opening sequence in the station, couples are dancing in front of a restaurant slate board advertising "Plats du jour" (suggestions of the day). Among these is "Boeuf bourgignon". The correct spelling should be "bourguignon".
When Isabella first hands the key to Hugo (at around 49 mins), you can see the automaton between them and there is no pen in its hand. Once Hugo winds it, and turns the key, the automaton has a pen in its hand ready to write, yet neither Hugo or Isabella placed it there.
When Inspector Dasté answer the phone from the bathtub, he answers it with his right hand all wet and soapy and the telephone is at his right side, next scene he's holding the phone with his left hand, the phone is behind him and both of his hands are completely dry.
After Hugo uses the tools to fix the wind-up mouse, he puts it on the counter. We see two tools next to the cup of tools on the counter. In the next shot, Georges Méliès is inspecting it and winds it up. Then, when he puts it on the counter to test it out, we see the tools are no longer in the way, though we did not see or hear them being moved.
A few pen strokes after the automaton begins to write, it stops and brings its elbow back to its side. In the next shot, however, the pen is still out in the middle of the paper, as if the arm were still extended. In the same scene, the automaton finishes its larger drawing, pulls its elbow back to its side, and raises its head upright to signal it is finished (at around 54 mins). In a subsequent shot, the head is still angled downward to the paper (at around 55 mins).
The concealed compartment in the armoire catches Hugo's eye because the right hand side of the bottom of the board covering it projects out rather than being flush with the rest of the front (0:59:19 to 0:59:20). In a close-up while Isabelle investigates it (0:59:51 to 0:59:52), it is the left hand side of the bottom of the board projecting out. She presses on it (0:59:53) and it slides into a flush fit, but, after a brief cut to Hugo, the left side is shown projecting out again.
When Georges Méliès confiscates Hugo's notebook, it's wrapped with a rubber band as Hugo places it on the counter (at around 5 mins). The next view shows it without a rubber band (at around 5 mins). Then it reappears on the notebook (at around 5 mins), and then, without being taken off, ends up on the table (at around 6 mins) when we first see the pages as Georges begins flipping through them.
At minute number 4, you see Hugo looking at Méliès' shop from above (from behind a #4 in a clock). As Méliès winds up the mouse and falls asleep, Hugo walks out from behind the walls without changing levels. He never goes down stairs to be at floor level.
After Isabelle starts moving the box out of the concealed compartment, it is sticking out a bit (as evidenced by its shadow) before a brief cut to the wobbly chair. In the next shot from above, the box is still within the edge of the compartment.
When Hugo follows Georges Méliès to his apartment, he appears to be walking close behind Georges, but in the cemetery (0:14:47 to 0:15:01 between tall statues) Hugo is shown by himself and there are no footprints ahead or around him either. Neither are there footprints when Hugo returns through the cemetery (0:17:15 to 0:17:20, a longer shot in dim light).
When Hugo is sitting in the armchair just before the automaton continues writing, Isabelle puts her right hand on the arm of the chair (00:52:53). A few seconds later (00:53:10) she has moved her hand farther up the chair arm and is leaning in a bit. The wide shot immediately following (00:53:13) shows Isabelle's right hand hanging at her side.
The Eiffel Tower was (and still is) the tallest structure in Paris. Yet when Hugo and Isabelle are at the top of the clock tower at the station, the camera clearly looks DOWN at the top of the tower (from 1:20:00 to 1:20:02 and 1:20:47 to 1:20:53).
The Montparnasse Train Station where most of the action is supposed to take place is alternatively shown on the same side of the river as the Eiffel tower, and on the other side. It actually is on the same side.
When Hugo and Isabelle are standing on a bridge, Notre Dame Cathedral is behind her to the west. The camera shifts to Hugo and he points behind himself (east) a mile away to the Montparnasse train station where he lives. The Montparnasse station is actually nearly two miles southwest of Notre Dame.
The old Montparnasse train station where the action takes place did not have a clock tower. The clock shown in the movie is instead reminiscent of the clock at another Paris train station, the Gare d'Orsay.
Georges Méliès was a real person, but the credits state that all persons in the film are fictional. However, this is not unusual for fictionalized stories even when there are characters based on real historical figures. The character is "fictional" in the sense that the things that character does and says within the film are not necessarily claimed to be actual actions and words the real person did.
Most of the many Georges Méliès excerpts seen in the film were made prior to 1910. Their accompanying piano music is the song "By the Waters of the Minnetonka" by Thurlow Lieurance, first published in 1913. But as these were silent films, and so would not have contained soundtracks, whatever music accompanied a screening of such a film would either be performed live or played from, say, a phonograph. So it is perfectly reasonable for this film to portray a older silent movie being screened with slightly newer music.
When the Automaton falls onto the tracks, the train has to break really fast to avoid it, but clearly the end of the track is 15 feet further down the track. So the train should not have been moving very fast in the first place. This was an intentional mistake; the scene was based on an actual derailment at this station, the same one featured in Hugo's dream, that was caused by a train coming into the station at too great a speed to stop in time.
When Hugo finds the key on the tracks in his dream, it is partly buried in the stones. In the next shot the key is on the sleeper (wood plank connecting the rails). But, since this is a dream sequence, continuity can easily change.
When the automaton is drawing the image, it begins by dipping the pen in an inkwell, and the nib emerges with black ink clearly seen on it. However, subsequent closeups of the pen show the nib dry, and a black pencil lead can be seen beneath the nib, which is what actually creates the marks on the paper.
When Georges Méliès winds up the toy mouse after Hugo fixes it, there are distinctive color changes in the pixels of the counter space beneath it to the right and bottom right of the mouse's path, revealing an editing clean-up gaffe.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
As Méliès relates his past (starting at 1:34:00), we learn that he built the Automaton before making his first film, in fact "using leftover pieces from the Automaton" to make his first movie camera (although the frames during this narration show him removing a part from the Automaton and placing it into his camera). However, when Hugo finally repairs the Automaton, it draws an image from Georges Méliès' most famous film A Trip to the Moon - It could well be that this sketch was imagined by Méliès before he actually made the idea into a film.