At approximately 31 min in, Rose breaks a glass when her alarm goes off. When the glass is shown at her feet she is wearing socks. A few seconds later as she is checking the door she is wearing lace-up shoes. Based on what happens later at a party later, there are several continuity issues that may not be "mistakes" but may actually be deliberate in order to show "breaks" in Rose's consciousness/sanity.
At one point Rose prepares a bowl of wet food for the cat, and in the next scene the bowl has dry food.
The viewers are supposed to believe that Rose's childhood home was simply abandoned after her mom died. This is ridiculous enough, but then light fixtures in the hallway (seen behind young Rose) no longer exist at the end.
At ~37:30Rose is wearing Bose headphones correct way round. Switches to closeup, they're backwards. Next shot, right way round again. A moment later, backwards again. Repeat.
Rose interviews a clearly psychotic patient alone in a closed room. The latter isn't restrained in any way, there's plenty of potentially dangerous items lying around, and the only way to call for help is a red phone on the wall on the far end of the room. The patient manages to slowly cut her own throat with a piece of a broken vase and security arrives after she had fully bled out. In reality, this would have been seen as negligence, both on the part of the doctor and the hospital.
After the traumatic experience, the hospital lets Rose work until she gets a nervous breakdown (or rather a hallucination caused by the Entity), and doesn't provide her with any counseling.
When Rose is reviewing the audio tape of her encounter with Laura, she increases the volume. Tones indicating a change in volume are heard and a graphical indicator shows it reaching its maximum level. A few seconds later, she somehow increases the volume further.