When Alita is playing motorball for the first time she knocks Tanji down before scoring. As she's skating away from the goal Tanji is in the background looking annoyed. In the next shot he is back on the ground, in the middle of the arena, where he fell.
The glove Shiren smashed the bug with changes position on the table in the different shots.
When Alita disarms Zapan in the bar, the sword bounces across the table and falls off the far side. But in the next shot, the sword is at her feet.
When Alita first wakes up in front of the doctor, he tells her his name. Once Alita adjusted to her new body, they went out and saw the sign outside of his shop which said his name and she asked him if that was his name.
When Alita cuts off Zapan's face, Zapan exclaims, "My face!" a few times. Ordinarily, those sounds (the 'm' and 'f' sounds) can't be made without his lips. However, since the character most likely speaks using a digital voice box of some kind (he has no lungs), he could still easily speak those words.
Alita does not know where she is, what the city in the sky is, but she knows the place she fought on was the Moon.
Alita had no hair when Ido found her yet when she wakes up her hair length and style is the same as it was in her flashbacks. If she's 300 years old, he wouldn't have known how her hair had looked back then. Her hair should have been different.
Just before the motorball tryout, Alita fails to recognize the Hunter-Warriors on the phony practice team despite meeting and fighting them during the bar scene. Only a couple of them were Hunter-warriors. The rest were bounties. They also weren't a phony practice team. They were low-level players in a qualifying match for the championship, that group specifically hired by Vector to kill Alita. Not to mention, Alita may have recognized them. But there's no reason she'd assume they were there to kill her.