After escaping Ratigan's trap, Basil changes from his sailor disguise to his regular clothes. When he's done, he is clearly wearing the blue sailor pants under his coat. When posing for the camera, however, he is wearing his regular brown pants.
After putting Ratigan's bell on a shelf, Basil leans back and crosses his arms. On the next shot, he is leaning forward with his right arm behind his back. Also, the bell disappears and is not seen again.
When Dr. Dawson and Olivia are seeking Basil's help at his house, Basil slouches down in his chair and starts playing the violin, holding it in his right hand and having the bow in his left. When Basil sits up in another shot, he is holding the violin in his left hand (like it's supposed to be) and the bow in his right.
When Basil and Dawson enter The Rat Trap, Dawson bumps into a poker-playing lady in green with a mole on her right cheek. As she turns around to blow smoke in the doctor's face, the mole is on her left; then when she turns back to the game laughing, the mole is back on her right.
When Ratigan's thugs are setting the spring on the overkill trap down, we cut to a close-up of Basil and Dawson tied down with the trigger-bar already set. But in the next shot, Fidget is setting the bar on the trigger.
During the "Let Me Be Good To You" scene, when Dr. Dawson dances on stage with the girl mice, the lead singer grabs him by his right arm and swings him around. Suddenly, she's holding his left arm and his head has turned completely around.
At the beginning of Ratigan's rant ("I have the power. I am supreme !", etc.) he snaps a crutch in his hands. He is still holding the two halves when we cut to a two-shot of him and the robot Queen, but when he jumps up on the banister on the next shot the two pieces are gone. (He is not seen holding them before the jump.)
When Ratigan's sanity snaps in the clock tower, he tears his cape in half to free himself. In his very next shot however, it is still there, albeit becoming much more ragged, only going down a little past his shoulders. But when he and Basil land on the hands of the clock, his cape, though torn, is back at its normal length.
Just after Dawson has said goodbye to Olivia and her father, there is a shot of him wearing his black overcoat but when it next cuts to him he says "It's time I was on my way too" and he puts the coat on.
In the mousetrap scene, the trigger mechanism is clearly above the ropes that tie Basil and Dawson. Trigger mechanism changes position between shots and is clearly under the ropes when Basil and Dawson trip it.
Although Disney's Dumbo wasn't released until 1941 and the film is set in 1897, the elephant bubble blower seen the toy shop is simply an elephant that happens to look like what Dumbo will. This is merely an easter egg for Disney fans. It is not meant to be a toy replica of Dumbo.
Fidget somehow finds Olivia at Basil's apartment, yet it's evident that Ratigan did not instruct him to go there. The arch-villain was surprised when Fidget told him he ran into Basil at the toy shop, and obviously didn't realize that Basil was already working on this particular case.
When Ratigan talks to Hiram when he is near the bottle, we see Olivia in the bottle, wearing her Tam O'Shanter hat. But Fidget wore it earlier (along with Olivia's scarf and dress), and we don't see Olivia's clothing after that until the final scene so Olivia couldn't have gotten her clothes back just yet.
At the end of the film, in the first two cuts (opening the door and close-up) of the Lady Mouse from Hampstead, she has a mole on her left. But in all the other scenes after that, the mole has vanished. Of course, her eyes are shaped differently, too.