Audio problem: While walking across the concrete garage floor Ryan's shoes make the sound you would hear while walking on hollow plywood.
Continuity mistake: When Driver is shown driving at the beginning, when it shows his hands, his gloves are on. But when the shot changes to show him still driving, his gloves are off.
Other mistake: In the pre-credit scenes when the chopper spots the runaway car, Driver guns it (he clenches his jaw, fists and you hear the engine revving and gears changing) but he never catches up with the two cars only a few feet in front of him (which are proceeding at normal speed) for the whole length of the bridge.
Continuity mistake: In this scene Ryan Gosling walks by a black mustang, you will see that his gloves are on his hands. In the shot, when he tries to break into the car, you still see his gloves on. But then moments later in the shot from his left, the gloves disappeared from his hands into his right back pocket. Though when you see him drive away, the gloves are back on his hands.
Continuity mistake: The first time Driver is taking Irene and Benicio home from the shop, Driver looks at her and asks her if she wants to see something. When he asks her, he is looking right at her. But the next shot, when she answers him, he is looking at the road.