Continuity mistake: After the pool incident when David is writing letters. Monica is looking over the letters that David has written. There's a point where you can see that she's still holding at least 2-3 sheets of paper but when the camera shifts back to the writing there's only one in her hand. There should be more than one paper in her hand. And when viewed from behind her back the sheets are neatly lying on top of each other, when seen from the front, the sheets are tilted.
Continuity mistake: When Monica first meets David, David walks out onto the carpeted floor and takes a step down to the wood tiled floor. He taps his feet on the wood tiled floor and then steps back up to the carpeted floor. He taps his feet again and turns around to face Monica. This is just before he tells her, "I like your floor." The mistake here is that Monica was pulled from the carpeted floor to the wood tiled floor before David walked in. So, David should not have had to turn around to face her because she would have been in front of him ... not behind him.
Plot hole: When Martin decides to antagonize David by eating spinach, he eats it from his plate. When they both eat the spinach frantically, they each eat out of a large bowl. Where did the second large bowl come from, and why would there even be a second one if they only need enough for three people?
Plot hole: Hair does not contain DNA so the boy's mother could not be recreated from just a snipping of hair. The hair follicle contains DNA but that would only be present if the hair had been ripped out. The robot bear that gives David the hair didn't really have a chance to get any hair as he left with David early on in the film and they never returned. Added to that even if he did have the opportunity, he wouldn't have thought about getting extra hair. They only find out about the DNA cloning at the very end of the film when the mother had passed away a long time ago.
Continuity mistake: They show David staring at the silver dancers dangling above his bed in which he sees his reflection. Next shot shows him down too low to be seeing his reflection. Also, there is a sound of a wind chime, however the silver dancers are not touching to make the sound.
Revealing mistake: When the robots start to scavange the pile of robot rubble the moon balloon comes over. One of the robots shouts, " Moon on the rise!". If you look carefully you can actually see the actors' real bottom lip behind a metal robot jaw that isn't meant to have any flesh or skin.