Piggy Piggy - S1-E6
Continuity mistake: The people that Tate murders at school have different wound patterns. Episode 5 they say what Tate did to them, episode 6 shows the attack. Episode 5 the goth girl says Tate asked her if she believed in God before shooting her - episode 6 he asked her nothing, instead just shot her. Episode 5, the guy in the letterman jacket illustrates his wound to be a single round to the head - episode 6 he shoots the letterman with a shotgun in the chest, not a handgun to the head. Episode 5 the cheerleader shows no apparent wounds of being shot - episode 6 Tate shoots her in the face with a shotgun.
Plot hole: It's been established that if you die on the property, you remain as a ghost for all of eternity. However, in Season 8, Moira's bones are retrieved and buried in a cemetery so that her ghost can escape and join her mother. However not all the ghost's bones are buried on the property, we know for sure that the Black Dahlia was not buried on the property and it's not likely Tate, Vivian, Ben or some of the others are buried on the property as well so they should not be tied to the house.
Suggested correction: Tate's victims from the school shooting are ghosts who can move between different locations. There's nothing to suggest that the ghost of the Black Dahlia isn't merely choosing to be in the Murder House rather than being tied to it. The same goes for Vivien and Ben, since Violet is going to stay in the house, so might they.
Afterbirth - S1-E12
Continuity mistake: When Stacy and Miguel (the new owners of the house), are making out in the kitchen, her shirt goes from off her shoulders to back on her shoulders across several shots.
Afterbirth - S1-E12
Continuity mistake: When Vivien and the maid are sitting in the kitchen, one shot shows the maid from the side with her hands in front of her on the table. The shot changes and instantly her hands are holding a coffee mug in front of her lips.