When Jerry and Penny board the double-decker bus in New York, Penny is holding an ice cream cone and Jerry's hands are empty. But in the next shot, as Penny gets her bus transfer, Jerry is holding the ice cream cone in his right hand even though we've never seen Penny hand it to him.
As Shirley lays back on the bed, she lays the teddy bear by her side with it's head towards her feet. A moment later, as Cooper tucks her in, the teddy bear's head is pointed is the same direction as her head, but neither he or Shirley have touched it.
When Pennie pulls the necklace out of the teddy bear's neck, she is holding it a few inches away from the bear, with all of her fingers around it. The camera then cuts to a different angle, and Pennie is holding it daintily between her thumb and pointer finger, very close to (perhaps touching against) the bear..
When Penny asks Jerry for his name, she mouths his answer at the same time he says it. Shirley Temple probably did this to stay synchronized for her next line.
The morning after Pennie finds the necklace hidden in her teddy bear, she is on her bed crying as Toni tries to get her to talk about what's troubling her. When Pennie explains, you can hear the audio of her saying "The necklace..." (right before she says "Daddy stole it and hid it in the teddy bear's neck") overlap with audio of her crying. Then, as the camera cuts from Toni to Pennie, you can tell that she did not say "the necklace" and that it was recorded later and added over the actual audio.