Plot hole: At the end of this episode Red takes off his shirt revealing his burned back, which leaves us to think that he might be Elizabeth's dad. However, this particular mark should have been noticed and written down in Red's prisoner file when he was taken into custody the first time (Episode 1). In that case Elizabeth would already know that Red is burned and could have asked some questions about it.
The Governor's Pleasure - S3-E1
Plot hole: The governor refuses to open the door for the prisoners when they have Vera hostage, and you can hear her saying "no" "return to your units" over the radio, which Channing had to overrule. Later on in the show somehow Vera doesn't know the governor refused and has to find out from Channing. But Vera was there next to the radio the whole time. They use normal radios without earpieces so there is no way she didn't hear. (00:38:05)
The Daughters of Jerusalem - S2-E6
Plot hole: Dinah Fortescue's fingerprints are supposed to be on a teacup but she's wearing gloves in the flashback. So how did they get there? (00:24:35 - 00:33:34)
Plot hole: Amy lets Scully place the watch in the break-room, but how could she have known that after Captain Holt steals the crown out of the locker, he would go there? And not, for example, go back through the window he came from.
Plot hole: It's revealed that the gym owned by Ray and Terry is a front for Ray to launder his income. This is why he doesn't want or can't sell it. Any person auditing that gym would know it doesn't bring enough income if the floors are in disrepair and the equipments garbage as Terry tells Pie when he gives him money to fix it up after Mickey pays him. Ray runs a consulting firm, he could launder millions through consulting, he has an office with employees. Get vending machines, anything that's cash. (00:42:00 - 00:44:20)