Revealing mistake: When Karg is raiding Kevin's house, we see a shot of a newspaper page. Karg's hook position allows us to see that the paragraph starting with "Many persons" is repeated.

Continuity mistake: Julie talks with her mother in the kitchen while her father stares. On his glasses, only one person is reflected. Uploaded picture has been zoomed to illustrate the mistake, but in no way is necessary when watching it on a screen.
Suggested correction: 1) The fact that you have to zoom in on the image invalidates the error. 2) In the image, I see a silhouetted figure against a bright light source. The second person could very well be to the side and therefore not visible.
Continuity mistake: Karg holds a newspaper with his left and right hands. In the immediate close-up, the right hand is gone. (00:51:47)

Continuity mistake: When a beast is about to hit Kevin, Evil-Lyn shows up, raises her hand with two fingers pointing upwards and commands to spare his life. A shot later, her hand is now pointing at him.

Continuity mistake: Lubic sees the pink car zoom by a tree and a bench. A second later, it passes by again.

Continuity mistake: When Kevin is cleaning up in the kitchen, he puts a load of Burger King related rubbish into the sink. In the next shot as Evil-Lyn enters the kitchen, the sink is now empty.
Continuity mistake: When the microwave explodes, the bucket of chicken disappears right after the explosion. (00:48:50)
Revealing mistake: The Eternia out-door scenes depicting the conquest of the Castle Grayskull area are actually one long shot subdivided into smaller parts. Look at the captured Eternian soldiers marched to the castle; they cover very little distance between the shots.
Continuity mistake: When Julie moves through the alley towards whom she thinks is her mother, in the first shot from her she is seen from behind as she walks to the wall on her left, and puts her hand out to it, almost as if to steady herself. The angle changes to a front view, and she is now standing next to a stack of crates, with her hand down by her side. The next time she is shown from behind, she is once again standing by a blank wall, with her hand outstretched to it, before she runs to her "mother".
Plot hole: When Gwildor tells Julie and Kevin that he can use the Cosmic Key to take them to any point in time they wish, Kevin asks just for them to go straight home. As Kevin and Julie go through the portal, Julie turns around and shouts to Gwildor to send them back before being dragged through the portal completely. When Julie wakes up in bed, she discovers that Gwildor has sent her back to the day her parents were killed. Julie never said where in time she wanted to go, so there's no way Gwildor could know to send her back to the day her parents died.
Revealing mistake: He-Man captures the hover platform and flies off to retrieve Gwildor's key. When he appears before his friends on the roof-top, the image used for him is much too flat, revealing its artificial nature.
Audio problem: The keyboard Kevin gets from the music shop seems to have an interesting trait, in that the high notes are on the left. This must either be bad "audio dubbing" or something to do with the shot perhaps being reversed, although I doubt this. It's most likely a post-production audio error as very very few keyboards or pianos are configured in this way.

Other mistake: When He-Man is fighting against the centurions on the floating disks, he tells the others to get inside a building. As he opens the door to the building, not only does it fall right off the hinges, but his sword also falls right off his back, pulling his cape off with it just as the scene ends. (01:09:20)
Revealing mistake: When He-Man is being whipped, on the last lash before the scene cuts to show the happenings on Earth, he reacts to the (animated) whip long before it hits him.
Revealing mistake: During the final fight scene, as He-Man attacks Blade with a rifle, Blade reacts to two hits that miss him.
Continuity mistake: In close up shots of He-Man on the flying disc, you can see some kind of safety support behind to stop him falling off, but in the far away shots, no such support exists.
Continuity mistake: The bald policeman who spends most of his time chasing Kevin, He-Man and the other Masters takes a shotgun from the music shop, cocks it, and then walks out. Upon sighting the Masters, he needlessly cocks it again, and then he cocks it for a third time before he actually shoots anything. He also has an apparently unlimited supply of ammunition for that gun - since it was not his in the first place, it is a cause for wonder where all his extra shells must come from.
Continuity mistake: In the scene where the mercenaries are introduced, in the first overhead shot, and in the close-up before their individual introduction, they are standing in line with Blade on the far left, followed by Saurod, Karg and Beast-Man. As Skeletor walks past them giving instructions, Karg and Blade change places.
Continuity mistake: In the first scene involving He-Man where he fights some of Skeletor's troopers and rescues Gwildor when two of them fire at him with RED lasers he deflects them back at the troopers but the lasers have turned BLUE. Duncan and Teela appear to help and their guns fire blue lasers but they never fire as He-Man has the situation under control.
Suggested correction: The blue shots are from Teela and Man at arms, as the next shot reveals. The actual mistake is that both shots come from the left and the 2 characters are on the right.
Continuity mistake: While He-Man is rescuing Gwildor, two of the black troops get hit by laser blasts coming from the left of the screen. These shots are supposedly from Man-at-Arms and Teela, both of whom are then revealed to be standing to the right.