(at around 7 mins) The band playing in the beginning has an upright bass-player, but the sound is clearly that of an electric bass-guitar. The playing style is also not true to an upright. The sliding note might give this impression, but the other notes are simply to light-touched to be an upright.
(at around 18 mins) Mr. Quick meets his death when the waving arm of the Shell Beach billboard chops his skullcap off, leaving the waving arm hanging to the left of the other arm. In following close-ups, the waving arm is missing.
(at around 1h 8 mins) When the doctor is pointing a gun at Murdoch and trying to give him the injection containing his memories, it is daylight outside through the windows behind them, but there is supposed to be no daylight.
(at around 12 mins) After John uses his 'tune' to force the Automat slot open to get his wallet, the broken spring and lever for the food slot fall in. When it cuts to a close-up of his hand taking the wallet, the broken spring is gone.
(at around 10 mins) During the time Murdoch is attempting to remember his first name, the little "Be Back at" sign in the haberdasher's window reveals one shot is a mirror-flip image, causing the sign's text and clock to appear backwards.
(at around 59 mins) When questioning a stranger on the rooftop, Murdoch falls and catches himself by grabbing the top of the railing on a fire escape below. In a following shot, he is climbing up the railing supports just below where he had caught himself.
(at around 17 mins) When Murdoch is on the Shell Beach billboard and collapses after reading the newspaper clippings he found in his coat pocket, a stranger appears/disappears/reappears on the steps in the background.