Visible crew/equipment: After Barney tells Andy he overheard Dr. Benson and Ellie "talking marriage" while spying on them, Andy heads over to the drugstore. Then, when Barney enters the jail cell, we can see a crew member's fingers (at the lower right edge of the bars, behind the wall) push the cell door closed. (00:19:45)
Revealing mistake: The woman in the foreground, as the siren is stuck, is carrying a paper grocery sack from "Hollywood Ranch Market" which was located on Vine St and Fountain Ave in Hollywood, Los Angeles. (00:09:35)
The Church Benefactors - S8-E20
Visible crew/equipment: At the start of the finance committee meeting, when Clara walks into the office modelling the choir robe, the shadow of the boom mic is visible on the door, in the top right corner of the screen. White tape marks are also visible on the floor, as Clara twirls in the choir robe. (00:08:15)
Continuity mistake: When Barney disguises himself as an old woman, in the very first shot he's wearing a dress with lace and tiny flowers, but in all the following shots it's an entirely different dress with large groups of flowers. (00:17:40)
Visible crew/equipment: When Barney and Andy are waiting for Virgil to arrive, just as the bus pulls up and stops, the reflection of the boom pole/mic is visible on the windshield. (00:03:50)
Aunt Bee and the Lecturer - S8-E10
Visible crew/equipment: When Andy parks the squad car in front of the house he asks Hubert about the picnic, and the reflection of the moving boom mic is visible on the windshield. (00:12:35)
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.