Deliberate mistake: When the Darlings discover the dead rat, they raise the curtain and cry out loud, but the rat is nowhere to be seen. Given Disney's long time history of refusals to show dead bodies or blood, it's highly possible that this was a deliberate mistake, but really awkward indeed.
Continuity mistake: From left to right, in the room where Tramp chases the rat there's a chest, a couch, a huge window with fallen curtains on a table. When the rat is killed the window disappears, and thus the curtains over the table. After aunt Sarah locks Tramp, the window reappears and the curtains are back on the table.
Continuity mistake: After the crocodile scene, Lady and Trump jump away and land on a narrow street inches away from the hyena cage. After the hyena stops laughing, the place is completely different: Wide and with a fountain in the middle. Lady is sitting still so they couldn't have gone somewhere else in between shots.