A Sentimental Journey - S1-E19
Revealing mistake: Dialogue in the episode specifically mentions the A2 and A282 roads near Dartford, Kent - the actual roads used were the A1 Barnet by-pass, Galley Lane and Well End Road. The error is revealed by the sight of a road sign that appears just as the Austin Princess Limousine turns on the the main road - the sign shows that the car is not turning on the the A2/A282 near Dartford, but the A1 at Barnet some 40 miles away.
Murder Ain't What It Used to Be! - S1-E7
Revealing mistake: Jeannie is not supposed to see Marty at all, but when she is in her flat and she is getting up off her sofa, she is seen walking around him, as if she can see him.
The Smile Behind the Veil - S1-E26
Revealing mistake: As Marty shouts at Jeff who is thrown into the river, he brushes against a bush moving its branches. Not something a ghost can do.
Other mistake: Kenneth Cope had to wear a toupee for his part. He did so, only it was initially on the wrong way, though no one pointed it out or noticed. If it ain't broke...
My Late Lamented Friend and Partner - S1-E1
Plot hole: When Jeff and Marty are parked, waiting for Sorrenson to exit the building opposite, Jeff says he should call the police. But he immediately follows Sorrenson and has no time to do so. Later, they arrive even though not called.