Claudia and the Secret Passage - S1-E8
Stupidity: The fact that Kristy found the ring that was missing for 40 years just sitting on a windowsill in the secret passage makes no sense; Dawn and Claudia were just by that window earlier in the episode in broad daylight.
Suggested correction: We don't actually see where Kristy found the ring; it was dark, and the girls were chaotic.
Stupidity: When Richie and Fonzie swap clothing and walk in for The Duel with the Frenchman (both with faces covered to conceal their plan), the whole audience watching The Duel gasp and are all surprised to learn they switched places. But, before the switch was revealed, Richie (as Fonzie) walked in with a nerdier style walk and Fonzie (as Richie) walked "cooler" and looked much more in shape. Also, the head covering Fonzie wore was sheer and you could still see his face, but everyone was STILL fooled.
Stupidity: The fact that the young mother in the hospital would give Janet (a total stranger to her) her baby to watch while visiting her husband in a different area is completely unreal. Nobody would really ever just give their baby to a stranger like that and leave. In addition, Janet took the baby into Jack's room for quite awhile.
It's Only Rock and Roll - S3-E11
Stupidity: When the gangsters show up at the house Pam and Dak Hampton are hiding in, Pam is captured. Dak then attempts to flee by running out the front door. What a dumb move. No surprise that he got captured as well.
Stupidity: Despite being a high-profile fugitive who was a local celebrity in his previous life, Saul does very little to alter his physical appearance, settling on just a mustache and glasses. Growing a beard, wearing a hairpiece, and altering his speaking voice around others would have gone a long way in preventing him from being made.
Stupidity: Al tells Peg to hide the meat and cheese tray from Bud and Kelly. Obviously you need to refrigerate meat and cheese yet Kelly says I swear I smell cold cuts. Do we have any? Al and Peg both say no. They both have to see it in the fridge every time they open it.
Stupidity: When Eddie finally finishes installing the VCR and it explodes, they get a man out to repair it. However, Eddie had previously told Richie that he'd stolen 60 VCRs; 17 in the living room and another 43 in the attic, so they could have used one of the other 59 and saved themselves a crate of Malibu.
There is Nothing Like a Nurse - S3-E10
Stupidity: Hawkeye and Trapper park a jeep over a foxhole to trap Maj. Burns in it. While it is certainly possible to trap somebody in a hole with, say, a sports car, an off-road vehicle like a jeep is particularly unsuited for the task - the chassis has so much ground clearance, even a quite fat man could squeeze out past it. You can even see the gap when the camera cuts to Major Burns screaming "Let me out!" (00:18:10)
Stupidity: When they are conferring before surgery everyone is keeping scrub except for Avery who has his arms crossed. Shortly afterwards you can see him look around at all his co-stars, realise he screwed up and try to casually raise his arms. (00:15:00)
First Kiss - S1-E2
Stupidity: When Jeff Malone leaves the sandwich shop, he leaves the sandwich on the table. It would have been very easy for them to tell what was on the sandwich without needing to wait for Max to get home to recreate it. (00:02:41)
Stupidity: Arashi tells Hoshino through his radio to call the HQ, but as shown several times throughout the show, his radio is perfectly able to do that directly himself. Even in this very episode later Hayata uses the standard issue radio to let the HQ hear the alien's requests. (00:07:20)
Always a Godmother, Never a God - S6-E4
Stupidity: Rory arrives in Star's Hollow for the baptism. She parks presumably near the church. When she gets out of the car, she does not have a purse or car keys in hand. She apparently left the car unlocked and left the keys in it. It's just plain dumb.