Jay accidentally makes a large hole in Hetty's bedroom wall. When he does this, there is no sign of the wood laths that should now be visible. Laths (tiny strips of wood that run horizontally across from wood beam to beam) were used in old homes to give something for plaster to adhere to so that a finished wall could be produced. No laths means modern day Sheetrock was used, which shouldn't be there.
When visiting the hospital, the chair and the objects on the doctor's desk clearly move when the ghost doctor leans into them, something that a ghost should not be able to do.