In the opening sequence there is a wheel of revolvers. Initially, there are four identical Taurus model 689, then right before the pistol fires that pistol becomes a Smith and Wesson model 681.
When Sherlock Holmes and Dr. Watson approach the initial crime scene, Sherlock's scarf, gray on one side and red on the other, had the gray side out and the red side against his skin. When we see them inside the brownstone, the scarf is showing only the red sided pattern.
The blood stains in the kitchen are too bright. Arterial blood is bright red when it spurts out of a wound but hours later, when Holmes and the police got there, it should have been dried up and dark, its color indistinguishable from venous blood.
When Sherlock and Watson get off the train, Watson's iPhone receives a call from Mom & Dad and clearly we see the options for "Answer" and "Decline", which would happen only if the iPhone was not screen locked, but after Watson selects "Decline", we hear the distinctive "keypad lock sound" of the iPhone, which would never happen unless the iPhone was screen-locked when the call is received.
When Holmes crashes Watson's car into the doctors car, the airbags don't deploy.
At the end of the show Holmes and Watson are watching a Mets-Reds baseball game on TV. This would be impossible since this story takes place sometime during the winter, as everyone is wearing heavy winter clothes and all of the trees are bare.
Sherlock makes a reference to Watson receiving a ticket at a cemetery at the corner of 86th and 3rd. None of the boroughs of NYC have a cemetery at such an address.
When walking around Saldua's apartment, every mug and cup, and the kettle all have their handles to the left, implying Saldua is left handed.
The pistol was in his right hand.
Sherlock should have spotted that.
Sherlock deduces that the pills in Saldua's bottle marked as Xanax was not, in fact, Xanax by asking Watson what the pill looks like, and she tells him "small, white, ovular." Different pharmacies use different shapes, colours, and markings for the same medication, and Watson, as a doctor, should know this.
Sherlock introduces Watson as his "valet", pronouncing as the Americans do - valay. A British person, especially one with the intellectual superiority of Sherlock Holmes would pronounce it correctly as valette with the t pronounced.