When Joey sets his sandwich down, it is toward the edge of the plate. When Minnie picks it up, it is near the middle and the crust is darker than before.
Just before Minnie runs into Michelle's room, Michelle is seen setting blocks on buckets. When Minnie comes in, Michelle is now sitting too far away from the buckets to have reached them. Then when Minnie runs into them, she knocks one bucket down. In the next shot, that bucket is standing upright again and there is a different assortment of blocks on the other one.
When Jesse discovers Minnie in his bed, a stain on his shirt disappears between shots.
When the girls lead Minnie and the puppies into the living room, one puppy can be seen walking at the rear. It disappears between shots.
At the beginning of the episode, when Joey takes a bite of his sandwich you can see how he meant to put it at the end of the plate, so it was easier for Minnie to eat it.
Danny tells Stephanie that he checked in on the local animal shelter in San Francisco and that nobody reported Minnie missing. However, in the beginning, Joey reads from Minnie's tag that she is from Ohio. It's likely that her owner's address and possibly a phone number is on there as well which negates the need to call the animal shelter. It's also unlikely that the owner would report Minnie missing to an animal shelter in California.
Minnie's dog tag says that she lives in Ohio, but Jesse later asks her owner if he can drive the girls to visit the puppies once in a while. This would be impossible given his family and work obligations and that the Ohio state border is 2,300 miles from San Francisco.