When Barnaby is in the workshop of Snape with the bicycle Helen Markham's car is already in the workshop. After Barnaby has left Helen is seen driving into the workshop with the damaged car.
When the intruder is trying to type the password on the keyboard, the number of key-presses does not correspond to the number of characters that appear on the screen.
When Mrs. Jeffers is leaving for her date, she is wearing a coat. When she returns, she is coatless.
After the event with the fire sprinklers, Barnaby talks to Jeffers in Jeffers' office for a full two minutes. During that time the clock on the wall keeps showing the same time, 9:50.
As described, the "kernel technology" would have evaporated into a cloud of molten silicon as soon as the on switch was pressed. The CPU power dissipation would have been thousands of watts. Think of the size cooling fans your computer's CPU requires, and how warm it gets ... now embed hundreds of CPUs in a mass of silicon. (as technobabble, it worked)
The executive says she is working late because she had to contact clients in California and she wanted to wait until they were awake. California is 8 hours earlier than Britain so a call placed at 4:00 pm in Britain would be 8:00 am in California, so it is not necessary to wait until late in the night.
Sprinkler systems operate in zones, so setting off the fire sprinkler in one room will not cause the sprinklers in another room to go off as happened in the show. In addition the fire alarm did not sound as it would in real life.
The bicycle mechanic tells Barnaby the chain broke because it was too tight. It is impossible to tighten a bicycle chain enough to break it. In fact, an old fashioned single speed bicycle should have no slack in the chain because there is no spring loaded derailleur to keep the chain tight enough to keep it from slipping off the gears.
The choir sings "Daisy Bell" with the line "...on a bicycle made for two". It should "built for two".
As Jones correctly points out, the Ford Escort was not made in 1963; it first appeared in 1967. However, this means that the car could not have had the pre-1963 registration plate as shown, with 3 numbers followed by 3 letters. A 1967 Escort would have had a number plate with 3 letters, 3 numbers and the suffix F, which signified registration of the car in late 1967.
Since the police know the car involved in the hit and run is dark green, it makes no sense to investigate people who do not own or have access to a dark green car. Barnaby's grilling of people who don't own green cars is illogical and a waste of time.
When Barnaby talks to Dr Bullard he mentions first degree murder murder, but English law doesn't divide crimes into degrees like that.
Barnaby and Jones are in the car after interviewing Melanie Jeffers, Barnaby mentions "George Jefferson" instead of "George Jeffers".
When Barnaby is talking to Dr Bullard about the car reversing back over the cycling victim at about 36 minutes he calls him "Joyce" rather than George.