It is established in the film that Russ is not capable of whistling, but when Wayne and Diane Szalinski lean over the grass towards the end of the movie and all of the kids scream, Russ whistles loudly.
(at around 53 mins) When the kids run and hide from the ant after he takes over their cookie, Amy had no cream filling on her mouth. In the next shot when she is telling Ron to be quiet, there is cream on her bottom lip.
From the window's side of Russell and Mae's bedroom, their bed is on the left side of their room, but when they are talking about little Russell, the bed is on the right side. Everything in the entire bedroom has switched sides from the first scene when Russell opens up his window to yell at Wayne.
(at around 28 mins) Wayne takes the kids outside in the garbage bag and puts them outside the fence. When the kids rip the garbage bag (possibly with the screw picked up in the attic) and crawl out, they are inside the yard.
At different points in the movie, characters are seen popping up in the middle of a Cheerios and riding an ant. However, most common species of North American ant that would be found in your average backyard would fit through a Cheerios with approximately the same amount of extra space as the character shown- even a Carpenter ant would be only moderately larger and would still fit through a Cheerios with room to spare. This would put the characters at about the same size as any given ants, not small enough to ride them.
(at around 21 mins) Szalinkski says that the Shrink Ray works by reducing the empty space in matter. If this is true, then the children's mass and weight would be exactly the same despite the reduced size. The trash bag Scalinkski would weigh several hundred pounds, the ant would not have been able to hold them, and the ground would have been compressing under the children's feet: the small surface area of their feet would mean that they would be exerting many tens of thousands of pounds of force per square inch.
During the scorpion/ant fight the scorpion is seen using his stinger multiple times not even striking the ant but the ground. Scorpions never waste their venom in random situations or even on prey that is smaller than themselves.
Szalinski says the shrink ray works by removing the space between molecules; however, a solid object has very little space between its molecules making it impossible to shrink as much as they are shrunk.
Nick's glasses are impractically durable. He has on more than one occasion while shrunk where they could've been broken in any way, shape and form, especially after being flung in the air not once but twice (from being on a runaway Bee and after nearly being chopped to death by a Lawnmower), his glasses would've had some kind of damage on them, like a scratched or cracked lens or warped out of shape.
(at around 2 mins) At the beginning of the movie, a letter carrier is delivering mail. There is a mailbox with the flag up. He should open it, remove the outgoing mail, put the flag down and deliver incoming mail. However, he just opens the box and delivers the incoming mail, leaving the flag up. While this is true, it is not uncommon for people to forget to raise or lower the flag accordingly.
After the kids are shrunk and Szalinski destroys his machine, he thoroughly sweeps the floor of the attic, ridding it of both the broken microchips and the kids. In spite of this, his shrunken thinking couch and chair are still on the attic floor undisturbed an hour or so later.
When the kids encounter the stream of dog urine, Russ suggests that if they could get some rope they could make a log bridge, only to realise they don't have any logs either; later however, after the scorpion attack, they conveniently have both to erect a fence to keep out any further intruders.
Szelinsky realizes at the end of the movie that the laser generating heat is what made his machine fail to work the way he wanted it to. But we see at the beginning of the movie that he uses a cold fluid (assumably liquid nitrogen) to try to maintain the heat inside the machine. So he should have known about this already.
In the second shot of the film where a mailman is seen walking and delivering mail for the Thompsons and the Szalinskis, you can clearly see the backlot wall in the background which is painted in blue to stand in for the sky.