When Holmes and Watson walk out of the house after seeing it for the first time, a police car is visible over Sherlock's right shoulder, but when Watson walks past the police car vanishes.
Dr. Watson's cane and his jacket shift positions in the restaurant scene as the shot changes.
Donovan holds up the police tape for John after he asks her where can he get a cab, she is seen lifting it clearly above head height. In the next shot it's much lower.
When Sherlock and Watson are in the restaurant, food shows up on the table even though they never ordered and a waiter is never shown delivering food.
When Sherlock and the cab driver take the pill, the cab driver holds it in his left hand, but after the cut his hand changes to the right one.
(I'm not sure if it's intended or a mistake. I couldn't find any official statement.)
Sherlock and Watson agree to meet at 7 p.m. to view the flat. When they get there, it's broad daylight. This is not quite possible in London unless it's summer, which, judging from the clothes they wear, is not.
Early on Sherlock is in the laboratory where he meets Watson for the first time. On the bench and on shelves are several flasks and bottles of blue and yellow colored chemical solutions. Factually 99.9% of chemical solutions are colorless and look like water. It's a common error in TV and movies to add dye or color to water to make the chemicals look "more interesting". The reality is a lab doesn't look as colorful as that. Several of these bottles of chemicals are lit from below or from the side to highlight them. This also isn't realistic.
Sherlock deduces that Jennifer Wilson is left handed, but when she reaches to the floor for the pill bottle, she is using her right hand. However Left handed people are perfectly capable of using their right hands and vice-versa.
When Sherlock gets in the cab, he asks the cabbie, "How did you find me?" but his lips don't move.
As Sherlock and John leave 221B Baker Street to catch a waiting cab, the camera crane used for the second part of the shot is reflected in the rear of the taxi as it departs.
When the CCTV of Brixton is being shown, a No. 14 bus with destination Putney Heath is visible. The route for this bus does not take in Brixton.
When Sherlock and John see the Taxi from the restaurant, supposedly in Northumberland Road, they run outside to chase the taxi up a road directly opposite the restaurant. When Sherlock then imagines the route the taxi will take the map starts on Wardour Street (not Northumberland Road) and there is no road directly opposite the start point as there should be.