When Arthur emerges from the muddy marsh, his entire head should be covered in mud. However, there is a clean outline around his mouth where he had obviously been breathing through a snorkel.
When Arthur Kipps (Daniel Radcliffe) first reaches Gifford Arms Inn in Sam Daily's (Ciarán Hinds) car, it is raining heavily and Arthur is shown with his clothes wet. When it cuts to the next scene where Arthur talks to the innkeeper Fisher (Shaun Dooley) at the reception counter, Arthur's clothes are suddenly dry. In the next scene still at the reception, his clothes are wet again.
The coin Kipps offers Keckwick for transport to the old house changes in size as the camera angles change.
At 54 min 40 secs, Arthur Kipps puts down his axe to tear wallpaper off. After tearing the paper off he steps back 2 or 3 steps and reads the inscription. The next shot shows him holding the axe again but he never bends down to pick it up.
(at around 16 mins) The ground level views show the carriage on a winding/curving causeway/road whilst the ground level shots show it on a straight one.
The muddy boy was so flexible and intact, even after being in the mud for all of those years. He would have been in a very decomposed condition and his bones no longer attached to one another. His clothing would have rotted away also.
When Arthur removes the lid of the coffin that contains the corpse of the woman in black, Sam does the Catholic sign of the cross (Father, Son, Holy Spirit), but mistakenly touches his right shoulder first rather than his left.
(around 1 hour) When Arthur spots the boy's footprints, they are much too neat and evenly spaced to be natural. They look like they were placed by a template.
Kipps travels by train, with antimacassars showing N&CR. This can only be Newcastle & Carlisle Railway, but this was absorbed into the NER in 1862, 40 years before the film's setting. Also, the locomotive is a Southern U class, which did not venture that far north.
When Kipps is looking through paperwork he finds Nathaniel's adoption certificate, dated in the 1800s. In the 19th century, the UK did not have a formal legal process for adoption as we understand it today. The concept of legal adoption only came into existence in the UK with the Adoption of Children Act 1926. The General Registry Office in the UK did not begin issuing Adoption Certificates until 01/01/1927.
In the last scene, the train has a post-grouping (1923) Class G headcode (light loco), but it should have a pre-grouping Class A (express passenger) headcode.
When Kipps is looking through paperwork and finds Nathaniel's adoption certificate, Jennet's name has the honorific "Ms." before it. Ms. as a title didn't exist in the 1880s, when the document was dated, and didn't even exist yet in the early 1900s when the movie takes place.
While Mr Jerome is initially trying to dissuade Kipps from staying in town, there is a point where Kipps tells Jerome, "I don't expect to be finished until Friday at least." Even seen from behind, it is clear his mouth continues to move after the sentence is finished.
(around 59 minutes) When Arthur goes outside to investigate the knocking on the door, a light simulating the lightning can be seen reflected in the window to the right.
When Mr. Daily leaves Arthur to Eel Marsh, he refers to Spider the dog as a girl. When someone tries to enter the house and Spider barks at the door, you can clearly see that Spider is actually a boy.