Chapter One: The Vanishing of Will Byers - S1-E1
Continuity mistake: When Will is being chased by the monster and runs into his garage, he pulls down his .22 rifle hanging on a wall rack. After he loads a few rounds aiming it at the door, the monster appears behind him. After Will and the monster disappear, the scene shows his rifle is back on the wall rack in its original spot. Did Will or the monster place it back on the wall rack while Will was being abducted? (00:07:25 - 00:08:10)
Chapter Two: A Touch of Evil - S1-E2
Continuity mistake: During the pep rally it is raining hard and everyone on the football field is soaking wet, but when Cheryl runs from the field into the locker room followed by Veronica and Betty, they are all completely dry and their hair is perfectly styled.
Continuity mistake: Christian is wearing a blue polo shirt, standing outside the luxury van in which Letty and Javier are being held captive. When he gets in the van, from that point on he is wearing a tan jacket over his shirt.
Episode #1.1 - S1-E1
Continuity mistake: The characters are looking at the Ten Little Soldiers. When one of them (playing the role of Blore) is banging on the glass, you can see the hands of the mystery writer ("Lombard") leaning against the base. He's leaning on the top of the display case in the next shot. (00:22:10)
Lost & Found - S1-E2
Continuity mistake: The way the shower curtain is draped over Todd's shoulder changes throughout the scene when he and Dirk are by the dumpster.