Supergirl positions her body twice in the flying ballet.
(at around 22 mins) [director's cut] When Selena holds on to the fridge door, an American flag appears and disappears between shots.
(at around 1h 35 mins) While trapped in the Phantom Zone, Supergirl falls into a pit of black slime that stains her whole costume. In the next scene the costume appears clean and laundered.
(at around 33 mins) The jogger in the first dorm scene runs through the glass doors twice.
In the "super showdown" scene, Supergirl reaches for the poles twice
(at around 40 mins) A hockey ball hits Linda's back and shatters into dust. This is meant to show how tough she is as Supergirl but granite, hardened steel, or even a huge diamond wouldn't shatter a ball it would just bounce off - while from air resistance the ball would be travelling more slowly than when the bat hit it, and yet neither shattered.
There is no explanation for how Supergirl would know her cousin is Superman with the alias of Clark Kent working for the Daily Planet. Nor is there any explanation of where her costume comes from. One must remember, however, that Kryptonian technology is thousands of years advanced from ours. It is not much of a leap of logic to surmise that they have been observing him through technological means of their own. As for the costume, the fabric is visible in the cushion of Zaltar's spaceship, which Kara uses to travel to Earth from Argo City.
In Superman (1978), Lois Lane says that her sister has three kids. In this movie, Lucy is portrayed as a teenager, in 1984 (six years later). Lois never specified that she had only one sister, let alone that her name was Lucy. Obviously she was referring to one of multiple sisters in so many words.
(at around 1h 30 mins) At one point during her journey to the Phantom Zone, only the upper half of Supergirl's body is contained within the crystal portal. This is not a goof. In Superman (1978) and Superman II (1980), only the faces of the three Kryptonian criminals are visible within the crystal portal. Therefore it is simply an abstract representation of each character's image, not a literal one.
Kara has to ride out of the micro-world before she gains her powers, but at the end of the movie she is able to fly all the way back home. It is plausible that her exposure to gravitational radiation during her journey from innerspace to outer space, as mentioned by Zaltar, altered her molecular structure to the point that receipt of her powers upon her exposure to our yellow sun was permanent, enabling her to fly back to Argo City unchanged. As her mother said, "Then she'll never be the same, ever."
(at around 23 mins) When Eddie the truck driver asks Supergirl if she's Superman's best friend, she replies "No, his cousin". It would seem at first that Supergirl knows her cousin only as 'Kal-El' and as she has only just arrived on Earth and she has no way of knowing yet that Kal-El is called 'Superman' on Earth. Therefore, she would not understand Eddie's question. However, one must remember that Kryptonian technology is thousands of years advanced from ours. It is not much of a leap of logic to surmise that they have been observing him through technological means of their own.
(at around 15 mins) After Supergirl lands on Earth, she flies very close to a lake, but she doesn't have a reflection in the water.
(at around 1h 18 mins) When Selena uses her magical powers to accelerate the fairground ride to super fast speed, the twinkling fairground lights in the background are also flashing much faster, indicating that it was filmed at a lower frame rate.
(at around 29 mins) As Supergirl changes to Linda Lee for the first time, she puts her right foot behind the tree, then picks up her left foot to bring it in front of her right, but before her left foot is out of sight, her right foot emerges from behind the tree.
(at around 1h 30 mins) When Supergirl is flying at super speed, the cars from the green-screen movie are also sped up.
(at around 1h 35 mins) When Supergirl is passing by the town, flying towards the mountain, she flies very close to the ground but no-one looks at her.
(at around 55 mins) When Ethan staggers around town in a daze, Jimmy Olsen comments: "He's probably on drugs - and he's gonna get creamed if he's not too careful." The last part of this line does not match his lip movements.
(at around 58 mins) During the "monster tractor" sequence, a boom operator is visible in the wrecked bar's mirror.
(at around 1h 9 mins) Crane used for flying reflected in the window when Supergirl flies out.
(at around 1h 9 mins) When the invisible monster crushes Supergirl against a steel door, she flies to fight him and the wires holding her are visible.
(at around 37 mins) When Selena visits the school, there's a tube reflected on her window. When her car starts to move we see it's part of a structure used for lighting.
(at around 1h) The tires sitting by the gas station during the construction machine running rampant sequence catch fire, yet there is no reason for then to have caught fire in the first place. All the sparks and fire were nowhere near the tires.