When Mrs. Banks bends to embrace the children (when the Constable brings the children back from the park), her head is turned to her left rather than looking at her children, obviously anticipating Mr. Banks' line which cuts off her embrace.
After all the chimney sweeps leave the house, Jane tells her father that every one of them shook his hand. This isn't, in fact, true. A big handful of them had simply tipped their hats to him on their way out.
In the opening credits, Mary is shown in a close up powdering her nose while sitting on a cloud. Her carpet bag is on her right and her umbrella is on her left. In a long shot, immediately after (which is animated) the carpet bag and umbrella are not present.
During the "Sister Suffragette" number, Mrs. Banks places a sash that says "Votes for Women" on Katie Nana. First the words are on her right side, next shot on her left side, and in the final shot, back on her right side.
Just before walking on the stairs made of smoke, they are walking single file on a rooftop in the following order: Mary, Jane, Michael, Bert. In the next close up shot, they are not in a line. They are in a group and Michael is in front of Jane.
At the end of Bert's performance as a one-man band, he hits himself on the head with a cymbal. However, immediately before this action, he is playing a concertina and afterwards there is no sign of the cymbal. Although he uses cymbals in his performance, they are not placed where he could so quickly grab one.
At the end of the movie, the Constable described the "missing" Mr. Banks as "about 6 foot 1". There is a common misconception outside of Europe that Britain only uses the metric system when in fact they use both. Britain didn't begin using the metric system until the early 1970's. Imperial measurement (slightly different to American) is still used almost exclusively for measuring people/clothing and for road speeds/distances.
In the penguin number Dick Van Dyke's pants change to short legs and long crutch to long legs and short crutch. This was deliberate, as Burt is clearly seen adjusting his trousers to match the shape/characteristics of the penguins he is dancing with just prior to and following the scene.
Obvious stand-ins for Mary, Bert, Jane, and Michael when Mary Poppins, Bert, and the children step onto a cloud which wafts them back down after exploring the rooftops (just before the "Step in Time" number).
The second time that all the Banks's furniture is moved by Admiral Boom's cannon, you can see a vase sitting on a column rock back and forth, and then be pulled back to its original position by a string.
During the "Supercali..." song, when Bert and Mary sing "The biggest word you ever heard and this is how it goes..." Mary's head becomes slightly transparent and you can see the animation behind her. This goof might only be on the remastered 2004 DVD version.
During the "I Love to Laugh" sequence, the strings attached to Michael are clearly visible in front of Mary's face when he first takes off. Later, the strings lifting Mary are visible when she joins the others at the table - showing that they are attached to her at the waist, just below her pocket.
When the constable brings the children home after Katie Nanna quits, Michael tries explaining why they ran off. While he is saying, "It was windy out," you can clearly see Jane mouthing that very line while he is saying it.
When Mr. Banks is dismissed from the bank, he takes Michael's two pennies from his pocket and these are seen as Edward VII pennies, which are correct for the period. However the coins are worn and battered, as if they had been in circulation for decades, whereas the longest an Edward VII coin could possibly have been in circulation in 1910 would have been eight years, as such the coins should have looked nearly (if not actually) new.
When Bert warns Mary Poppins that Uncle Albert's problem is contagious, just before Mary opens the doors to the tea parlor and they all see him laughing on the ceiling, Jane asks "Should we get spots?" This line doesn't make sense. Obviously, Jane is concerned about getting sick, and should have asked, "Will we get spots?" or perhaps she could have said "Will we get shots?" Mary Poppins answers back, "Oh highly unlikely."