Continuity mistake: When Shifu performs the Wuxi finger hold on Po, in one shot, we see all of Po's fingers, except the one Shifu is holding, are closed to his palm. In the next shot, all his fingers are open. (00:22:05)
Suggested correction: If you slow down before the part where the camera pans onto Po when he's surprised, his fingers open.
Continuity mistake: When Tai Lung is at prison, and he breaks his shield-thing off, the guardians fire four arrows at him and he evades all of them. He uses the fourth and last one to break the shackle off his hand. But in the next camera angle, there are five arrows stuck in the floor. Where did the fifth one come from?
Suggested correction: The fifth arrow appears after a space of 17 seconds. During this space, not only a fifth arrow has appeared, two other crossbows are rearmed and ready to fire. It is plausible to assume the guards have done all this. This is the natural progress of the film.
Continuity mistake: When Po's Dad puts the apron on his following their hug the apron appears out of nowhere, having not appeared in any previous scene.
Suggested correction: If you are talking about the event at 1:07:40, it is not a mistake; Po's stepdad put the apron on him. There is a suspicious woosh sound and Po's right eye (left side of the screen) opens in surprise. The apron also unfurls, as if it was just worn.
The mistake seems valid. When Po's dad see him, he raises his arms (wings) and his hands are empty. When he hugs Po, his arms in no way come close to wrapping around Po's belly. So why we can't see what Mr. Ping is doing during the hug, his position doesn't move, so there's no way he could tie the apron around Po if we assume he somehow had the apron in his pocket (even though there's no evidence for that).
I know. But the whoosh version that the filmmakers have chosen is more fun than your literal-minded version. It is an established tradition in cartoons to replace very fast actions with cartoonish materialization accompanied by a whoosh sound. Following that tradition is not a mistake, especially when the film is a work of fantasy.
Other mistake: When the villagers return to their homes, Po is seen emerging from an alley obscured with dust. In the shot where we can only see his profile, Po's "cape" is blowing in the wind, but the dust does not seem to be affected.
Suggested correction: You'd have been correct if the only inconsistency was a misaligned cloak. But that scene is far too elaborate. At first, we see a stereotypical legendary warrior in a long cape and straw hat pacing through magical dust. But soon, Po's figure transforms completely; even the dust vanishes without leaving a trace. I think what we see is either what the villagers thought to have seen or a mirage that the magic dust had cast.