I usually notice things like. Wrecked cars in that suddenly have no dents in them. Commando is a good example, Arnold rolls a car over on its side gets out, flips it back over, the drivers side is crushed in , the scene edits and you see him drive off in a brand new car.
Or also i notice stuff like in Star Trek TNG. they are all in the ready room for instance with their nice glass of what is assumed to be water ...and you'll notice the liquid level in the glasses changes constantly, full empty, half full, ect. Specially with Jonathan Franks, he drank alot of the water between takes and during some scenes, so the prop guys were always having to refill it,but never got it to the same level twice it seems.
Theres also things that one might notice like.. In the new BSG, you'll see in one episode ( i cannot remember which one) the Ncc-1701 in the fleet, and later in the same episode you'll see a flrefly class freighter in the fleet as well.
and i think someone has pointed out a millennium falcon and a few other sw ships in some ST movies.