Communication Problems (a.k.a. Theft) - S2-E1
Plot hole: Basil torments Mrs. Richards by miming when he is speaking to her, so she will think her hearing aid has failed. However, she speaks twice during the scene, and she would hear herself - obviously - and would know that he was trying to fool her.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Revealing mistake: In the scene where Basil and Manuel have put the corpse in the room behind the reception, and the Major comes over and speaks to Basil, the actor playing the corpse blinks.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Character mistake: When Sybil is telling Basil that the two men and the woman are coming to collect Mr Leeman but he's already pronounced dead, she tells her husband that they are coming to collect him "at lunchtime". However, his colleagues told Mr Leeman they'd collect him at 9:30 and not lunchtime.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Revealing mistake: Just after Polly comes down to to the kitchen to tell Sybil that Mr Leeman's dead and they leave, Basil wanders back in, to learn the news. Watch the wall on the right on the staircase when Basil shoves past Sybil on the stairs. The wall shakes so much at that point it's virtually impossible to miss.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Visible crew/equipment: In the kitchen scene with Basil, Sybil and chef, Basil says, "Would you believe it, I give him his breakfast, I take it all the way upstairs." At this point, the shadow of a boom microphone on the wall to the right is visible.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Revealing mistake: When Basil, Manuel and Polly are trying to hide Mr. Leeman from the other guests, they put him in a cupboard. While they are talking to the people staying in the room, Mr. Leeman's hand comes out of the cupboard. Watch carefully, and you can see the actor opening the door from the inside, and then putting his arm down, making him look like the corpse.
Communication Problems (a.k.a. Theft) - S2-E1
Visible crew/equipment: In the scene where Polly tells Manuel to go and fetch toilet paper for Mrs Richards, the boom microphone shows up at the top left for a few seconds.
Visible crew/equipment: When the Hamiltons arrive, Mrs Hamilton says 'May I introduce my husband.' The boom mic is seen at the top of the screen.
Continuity mistake: When Basil is talking to Mr. Hamilton in the dining room there is smoke coming from the kitchen. The next moment he goes into the kitchen and all the smoke is gone. (00:24:00 - 00:25:00)
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Visible crew/equipment: When Mr. White asks if Basil can clean their room later the boom mike shadow is visible on the door behind Mr. White.
Visible crew/equipment: When Basil's trying to persuade Sybil about the woman in the guest's room, the shadow of a boom mic appears on the top wall.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Continuity mistake: When Basil, Polly and Manuel are trying to hide the body in the cupboard in the guests' bedroom, the door behind them leading to the bathroom mysteriously opens and closes between shots throughout the scene.
Visible crew/equipment: After Mr. Johnson refers to the Torquay guide as the 'world's shortest book' and leaves the hotel, you see a boom mic coming in at the top of the screen for about 3 seconds.
Basil the Rat (a.k.a. Rats) - S2-E6
Revealing mistake: When Polly and Manuel are walking down Fawlty Towers drive with Basil the Rat in the box, there is an arch over the front entrance with Wooburn Grange Country Club written on it. This is the name off the building used as the location for the exterior scenes of Fawlty Towers; this is most visible on the DVD.
Basil the Rat (a.k.a. Rats) - S2-E6
Continuity mistake: When Basil and Sybil are talking to the public health inspector in the kitchen (on his second visit), Basil breaks two bottles of wine. When he removes the second bottle, there are two bottles remaining on the rack, but when the camera cuts away and then back when he breaks it, there is only one bottle left on the rack.
Revealing mistake: When Basil is climbing up the ladder to view the inside of Mr Johnson's room, he accidently peers inside Dr Abbott's room. When he starts doing his "checking" routine on the window before he falls backwards, the pane flexes, indicating that it's made of clear perspex.
Visible crew/equipment: When they are in to see Sybil, as they switch on the light, the boom mike is visible on top of the screen.
Revealing mistake: There are noticable disparaties between the locations of rooms and the windows on the outside of the building. This is particularly noticable in this episode when Basil climbs a ladder to look through the window.
The Kipper and the Corpse (a.k.a. Death) - S2-E4
Continuity mistake: When the guest from number 8 is going up, in one shot Basil is looking on his watch when you see him from behind but not from the front a split second later.
Continuity mistake: Near the beginning, when Sybil goes into the office after speaking to Basil about the date, the lights are obviously on in the office, but when she walks out, it can just be seen on the right that the office is in complete darkness, and then when the shot changes back to the regular one in the lobby, the office is lit up again, without anyone going in there in the meantime.
Answer: It's "Leisure Complex" by Dave Gold and Gordon Rees. Can be found on YouTube. The relevant clip is around 25 seconds in.