One somewhat funny example of this is in the first ST:TNG episode "Encounter at Farpoint". In one shot, the captain asks Data a question, and the camera turns to him to show him standing from his seat at the conn and answering. At the bottom of the screen, it's plainly visible (in the new Blu-Ray version) that a patch of extra carpet is under the edge of the seat. It was probably put there to level the seat or something. At the time, this was ignored, because on a standard SDTV screen, the edges are all rounded, so the very edge of the frame isn't normally visible.

Another thing that's plainly obvious in TNG's remastered version is all the black cardboard placed over the display screens in the back of the bridge, to block glare from lights. In SDTV, this wasn't noticeable because the quality was so bad.