Revealing mistake: When all four actors are up on the rooftops, look closely at them when they all step onto the smoke platform that brings them down. All four actors are doubles. (01:46:50)
Revealing mistake: An early scene with the constable and the children, Jane silently mouths along with one of Michael's lines.
Revealing mistake: When Mary Poppins takes the children on an outing to the park, look at the front steps when the front door opens. They are not steps but flat ground painted to look like steps. (00:36:52)
Revealing mistake: In the scene where Mary Poppins is removing the large plant from her carpet bag, Michael inspects the underside of the table, where the table leg casts a shadow across his face. When Mary crosses in front of him (still under the table) she does NOT cast a shadow, although she should.
Revealing mistake: At the very beginning of the movie, when Mary Poppins is sitting on the cloud, you can see the string attached to her bag just as it almost falls through the cloud.
Revealing mistake: There is a certain point in the song "Step In Time" (I believe it is when we see a couple of men switching their feet on the top of a chimney) where against the sky you can see where a wall and ceiling meet. This is because the scene was shot indoors.
Revealing mistake: During the close-up shots of the ceiling tea party, Bert and Uncle Albert have shadows on the wall directly behind them. However in most of the wider shots, (which have obviously been shot against a blue or green screen) the shadows are now gone.
Revealing mistake: In the tea party on the ceiling sequence, 2 shots after Uncle Albert tells his long underwear joke, Jane does a backflip in the air. If you look closely you can see the imprints that wires that she was hanging from made on her dress. (01:15:55)
Revealing mistake: When the penguins walk in front of the table where Mary Poppins and Bert are sitting, the table can still be seen through the penguins. (00:48:30)
Revealing mistake: During the 'Spoonful of Sugar' musical number, when Mary Poppins is singing to the robin in the children's room, the wire connected to the audio-animatronic bird can be seen just under Poppins' two fingers, very close to her palm.
Revealing mistake: After the credits all the smoke coming out of the chimneys remains still, revealing it's a painted set.
Revealing mistake: When Mary Poppins has the robin on her finger, just after the outside shot where the robin does his whistle bit, you can see the cables used to operate the bird on the back of her hand while she turns away from the window.
Revealing mistake: When the rooftop dance sequence begins and 3 of Bert's pals appear to shoot out of 3 round smoke stacks, they are actually coming from behind the stacks. (01:47:35)
Revealing mistake: Mary takes a huge mirror out of her bag and the kids stare in awe. Problem is that Michael's view points in a different direction, giving away that they were looking at the void and Mary was a superimposed image to achieve the effect.
Revealing mistake: In the tea party on the ceiling scene when Mary takes out her watch, watch closely, she doesn't take it from behind her jacket. When she seems to reach behind her jacket it just appears in her hand. When she puts it back she reaches behind her jacket again, but it just disappears just before she puts her hand back there.
Revealing mistake: In the scene with the queue of Nannies, and the chimneytop scenes, the wires holding the actors can be seen clearly.
Revealing mistake: When Mary is taking the things out of her bag when she arrives, she takes out a large floor lamp. You can see the lamp's post coming up through the bag and the table as she pulls it out.
Revealing mistake: As the squirrels play around Bert and Mary, the background can be seen through their tails. (00:45:25)
Revealing mistake: As the sweeps dance and jump from building to building, the edges of the matted in street that hides the sound stage floor is easily discernible.
Revealing mistake: In the scene with the three chimneys, the sweep coming out of the middle chimney can be seen actually coming up from behind it, as his arms going up are visible.
Chosen answer: They recorded her singing in two separate parts and fixed them together. The editors did a good job on it! It may have been made a long time ago but even then the technology was quite advanced enough to carry out that scene.