Plot hole: In the season 1 finale, after they have brought James into the the station Cooper hands Truman a note and asked if he would investigate this. A few minutes later Truman comes back in with the coke that Bobby planted in James' gas tank. Cooper then asks James why he had coke in his gas tank. Cooper shouldn't know where the coke was found, Bobby never mentioned to Lucy where he put it and Truman never told Cooper where he found it.
Continuity mistake: When Pete Martell finds Laura Palmer's body, there is a piece of wood on the ground near Laura's head as he approaches her. It disappears between shots. (00:04:20)
Continuity mistake: When Lucy is badly explaining to Sheriff Harry Truman which phone she transferred the call to, in the window reflection, you can see the Sheriff moving around the office to find the phone. But in the close-up cutaways of Harry, he is standing in one place. (00:05:07)
Visible crew/equipment: When Sheriff Harry Truman takes the call from Pete Martell at the police station, a reflection of a seated crew member wearing headphones (it actually looks very much like David Lynch!) is visible in the glass divider of the office the Sheriff enters to answer the phone. Clear as day. (00:05:13)
Visible crew/equipment: When Sheriff Harry S. Truman first leaves the station, film crew are visible as they dolly back across the Ford Bronco. (00:05:55)
Other mistake: When we are first introduced to Leland Palmer and Benjamin Horne, they are discussing a contract in front of a large, lit fireplace. A couch (on which Leland is seated) is in front of the fireplace. Strangely, the flames of the fire are visible through the couch, and also Horne's legs are at one point. (00:11:09)
Visible crew/equipment: At the diner, when Bobby asks Shelly if she needs a ride home, the reflection of a boom microphone is visible in the horizontal glass mirror behind Bobby (far left of the screen). (00:16:15)
Visible crew/equipment: When Josie and Catherine are arguing at the mill, just as Pete says, "Oh Catherine, please," he moves toward the chair, and the reflection of a crew member wearing a cap is visible on the window, at the left side of the screen. (00:32:25)
Visible crew/equipment: When Bobby is released by the police, he speaks to his friend in the hallway of the station about the possible identity of Laura Palmer's killer. To avoid eavesdropping, Bobby tells his friend that they should speak outside. As they walk through the glass doors towards the exit, Bobby encounters his father and two other people. Tape markers are visible on the floor placed at their feet. (00:55:53)
Visible crew/equipment: While Truman and Cooper are at the bank, when they take the safety deposit box and walk over to the table, there's a green tape mark on the floor by Truman's feet. (01:03:20)
Continuity mistake: During the barroom brawl, Bobby and Mike start fighting and Donna escapes out the back exit with one of the bikers. They leave on his motorcycle to meet James. Dale and Sheriff Harry are trailing them. Then a cut back to the barroom and Bobby and Mike are still fighting the bikers. However, if you look in the top right of the shot, you can see Donna and the biker still behind the bar, trying to leave. (01:19:13 - 01:19:50)
Visible crew/equipment: At the very end, Sarah is alone while she's lying down, but when she has the sudden vision of Laura's necklace being dug up, the reflection of a crew member is visible in the round mirror. (01:32:50)
Visible crew/equipment: Donna overhears her father and mother discussing Laura's golden heart necklace and the picnic video. Then, just as Donna enters the bedroom, there's a white T-mark on the floor at the bottom of the screen.
Continuity mistake: The credit has "Lucy Morgan." Later, her name becomes "Moran"; see the nameplate on the desk.
Episode Two - S1-E2
Visible crew/equipment: When Audrey spots Cooper ordering breakfast, her yellow tape mark is visible in the doorway at her feet. (00:04:14)
Episode Two - S1-E2
Visible crew/equipment: Just before Norma bumps into Nadine inside the store, there is a shot that follows a model train running on a track. At one point, a crew member wearing a white shirt is reflected in a mirrored panel in the store's main window. (00:21:12)
Episode Two - S1-E2
Visible crew/equipment: When Deputy Hawk is interviewing the parents at the hospital, there is a square window behind them. The camera is reflected in it as it moves in towards them. (00:32:39)
Continuity mistake: At about 23:21, Dale Cooper is writing on a chalkboard that has the letters R and T on it, not circled. Moments later, at 23:30, the letters are circled, even though Cooper has not marked the board. (00:23:20)
Suggested correction: Actually, there is a cutaway shot to the two policemen, and when the shot changes back to Cooper (with the letters now circled, as you say), he does have the chalk in his hand and then circles the "J." I think it's implied he circled the R and T in the cutaway. There was enough time.
Episode Four - S1-E4
Visible crew/equipment: After the court hearing scene, Cooper and company drive up to the veterinary clinic. Just before they arrive, we see a stationary shot of two petrol pumps. In the glass display on the pump on the left-hand side, a crew member is reflected. Cooper's vehicle then drives into view, and the same crew member is reflected again on the car door. (00:19:49)
Episode Four - S1-E4
Visible crew/equipment: When James is on the phone to Donna, Madeleine walks into the diner and stands at the counter. As the camera pans from James to show this, a white tape marker is visible on the floor near Madeleine in the right-hand corner of the shot. (00:31:16)