When the dog cuts the hole in the barn floor, technically, when he returns down the hole ("and for God's sake, don't forget to wag those tails") the cut out circle should have fallen back down as there was nothing holding it in place.
When Scotty is kicking the soccer ball, after tryouts, in the back yard, at first his net is square with an ex in the middle of it placed next to a wall. When the shot is panned out, the net is circular with no ex and is in an open part of the yard.
When the cats pull up to the gate of the flocking company in the limo, there are two men standing and talking on the other side. A moment later, when the limo drives through they are nowhere to be seen.
In the first scene of the movie, the Brody's car pulls up in front of the house. As it does this, a jogger jogs up to the car and is level with the front of it when the shot changes. The next shot shows the car stopping at the side of the road, and the jogger jogs up to the car again.
In the Pie Mom's house, Buddy runs upstairs following the cat and turns corner. We get a closeup of the dog's eyes first, followed by a wide shot of the hall; the cat is sitting at the end of the hall. In the very next close-up shot, the cat is up on all fours. The power cord of the table lamp also changes position between shots.
When Mr. Tinkles is first introduced, the cats he is speaking to hide under the table when the maid suddenly comes into the room. However in the next shot when the table is shown again, the other cats are nowhere to be seen. They then reappear when Mr. Tinkles is being carried out of the room by the maid.
When the Russian Blue throws the spike bomb at Lou and Butch, it explodes, wrecking the entry hall and the entire living room beyond. However, the spikes were geared in a circular fashion, meaning they launched in every direction, not just the hall and living room. The dining area behind the cat is completely undamaged, and during changes in the viewpoint, parts of the living room flip back and forth between damaged and repaired.
When the dog agents are trying to break into Professor Brody's Lab, the camera shot changes to the dog's point of view. He explains that he doesn't know which wire to cut (as all of his vision is in black and white), but dogs do not see in black and white - their only visual impairment is that they see in the same way as a human with color blindness.
When the cats drive into the factory you can see the tire marks from previous takes. It is extremely unlikely that such marks would be caused by normal slower moving traffic, especially considering the erratic route.
When Butch is complaining about the mix up with Lou, he says that Lou is so young that he still has his "you know whats", implying that Lou has not yet been neutered. However, the dog playing Lou is clearly neutered.