Plot hole: Doyle attends Gondorff's betting shop three times, and he listens to the announcer calling three races from three different race tracks - Narragansett in Rhode Island, Belmont in New York, and Riverside Park in Missouri. He cannot possibly miss the fact that the same announcer calls all three races! J.J. Singleton, the race caller, has an instantly recognisable voice, and Doyle wouldn't be fooled for a second. Each race track would have had its own announcer.
Plot hole: Edward Lionheart severs the head of theater critic Horace Sprout while Sprout and his wife are asleep. (Lionheart gave them both a knockout injection). Sprout's wife (and his maid)discover the severed head in the morning, and both scream and faint. But in the next scene, critic Perry Devlin retrieves his morning newspaper and then reaches for his bottle of milk, only to find Sprout's head perched on the milk bottle. How did the head get from Sprout's house to Devlin's place?
Plot hole: Webster meets Laura at a party and she takes him to her house - it's not really her house - and says he has to break in to show off his talents, but when Webster does break in, he walks into a pitch-black house and he walks straight to the light switch and turns on the lights, so Laura can come in to a lighted house. Problem is: Webster has never been to this house before, it's pitch-black inside (it's so dark we can't see Webster at all), and the light switch is not where most switches are - on the wall next to the door - it's on a pillar, which is really far away from the front door; there's no way he'd be able to go straight to it.
Suggested correction: There are a lot of things Doyle is processing when he's in the shop. There's a lot happening, and he has a lot on his mind, and increasing pressure and stress each time. It's quite possible that he wouldn't notice the accent of the announcer, something he has no reason to doubt.
Rubbish. During his first two visits, he sits quietly listening to the race announcements. On his second visit, he would recognize J.J. Singleton's distinctive voice and would realize something was very wrong.