When having a beer with Constable Strange in the pub, Endeavour Morse finishes his pint. In the next shot he has at least an inch and a half left in his glass.
Pamela Sloan has a savings book & letter from Dr. Cartwright addressed to 173 Noel Road. When Endeavour visits her the number on the blue front door is No.17.
In the story's opening scenes, one of the cast is seen at the Post Office handing over a ten shilling note to buy a postal order of the same value. However, postal orders always cost slightly more than their face value, as costs of a few pence were added on for processing.
Morse finds a bottle of pills named digoxin near the girl who has a heart attack and asks her doctor what they are for and the doctor says they are "deadly nightshade" or "digitalis" for her heart problem. Digitalis is actually derived from foxgloves and deadly nightshade is the basis of "bella donna" which is a poison, hence its name.
Morse is given a bag of coins from gas meters and told to "count the shillings". However the coins which he is counting are far larger than shillings. The one seen in close up is a crown (5 shillings) and all the others are the same size so must be crowns too. Although crowns existed they were only issued as special commemorative coins (e.g. the Churchill Crown) and were not in general circulation. They wouldn't have fitted in a gas meter.
In the crossword puzzle that Endeavour helps Rev. Monkford complete, the numbering is wrong - they go 24-26-27-28-29-30-31-25-32-33.
The show is ostensibly set in 1965, however a shot of a poster of the Periodic Table of Elements includes Nobelium which was discovered in 1966. (The poster does not include Rutherfordium, discovered in 1969, setting a latest date it could have been printed.)
At about 19:06, Morse's record of Rosalind Calloway is seen WITHOUT her signature -- she signed it in the pilot and her signature is clearly visible at about 1:19:52 in the pilot.
During a shot of the Sheldonian Theatre, a bust of one of the Roman emperors is shown. This particular bust was carved between 1970 and 1972, and so it is not have been possible for it to be there in the 1960s.
When Morse asks Dr Prentice about digoxin and digitalis, Dr Prentice replies, "They don't call it deadly nightshade for nothing." This is wrong: the deadly nightshade (belladonna) plant produces atropine, not digitalis which is derived from the foxglove.