Continuity mistake: When Howard's wife is leaving Joel's apartment, the first two shots show her trench coat belt hanging from the bottom of the driver's door. In the third shot the belt is no longer hanging from the door. (01:19:30)
Visible crew/equipment: In the beginning, when Joel "first" goes to Kate's apartment for drinks and Kate is discussing her worry of wasting time, when the shot switches from a side shot to a straight shot, a boom mike shadow moves above Kate's head.
Continuity mistake: Mary and Stan are lying on Joel's bed talking, with the plate of food between them. When the camera looks down on them, Joel's hand is on the plate. The shot changes at one point to a close up of Mary's face and Joel's hand is not there anymore, but reappears when the shot changes again. (00:49:20)
Continuity mistake: When Joel gets out of bed at the beginning, he flips back the corner when sitting up. In the next shot, he kicks off the comforter, but the corner is back over him again. (00:00:50)
Continuity mistake: When Jim and Kate are talking on the train near the start, there is a point halfway through a sentence where Kate's arms go from being folded, to being straight in front of her - instantly. (00:05:55)
Continuity mistake: When Mary is on the street talking to Howard, her hair is quite dishevelled but as the shot goes to her face it is suddenly smooth and tidy, as though just combed. (01:20:55)
Continuity mistake: When Mary and Stan are lying on Joel's bed and smoking pot, Stan is talking about The Clash and Mary is talking about how great their boss is. They are sharing a single joint. (The point is shown being passed back and forth.) However, Mary passes it to Stan. Then the camera focuses on Mary, who is now holding a joint.
Continuity mistake: Just before Joel walks in to see himself in the chair for the first time, he has a digital blood pressure monitor velcro'ed to his arm. Before he walks into the other room, the end of the strap is on the side of his arm facing away from his body. When the doctor takes it off, he has to reach around to the back of his arm to grab the strap. (00:33:00)
Continuity mistake: When Clementine is getting ready to go out, Joel spreads ketchup across his neck and lies on the floor. Shots later, when Clementine has left, Joel sits on the couch and wipes it off, only there's no ketchup on his neck. Instead he wipes it off from his lips and cheek.
Continuity mistake: When Clementine and baby Joel are in the kitchen, Joel's yo-yo and crayon continue to switch places between shots, throughout the whole scene.
Continuity mistake: When we see Joel in his car, the blue dot on his right temple is large and close to the corner of his eye. When we see Joel a moment later in the lobby, the dot is now much smaller and is further away from his eye.
Visible crew/equipment: When Clementine storms out just before the bit where Joel chases her down the sidewalk he gets in his car. The shot from beside the car shows at least two shadows of the crew moving around on the car.
Visible crew/equipment: When Joel awakes he looks at his dented car. When he walks in between the cars to look at the back of his own car, you can see the reflection of a boom mic being poked round to pick up his footsteps.
Continuity mistake: When Joel first meets Clementine on the beach, the close up shot of her shows her feet in the water. In the next shot near Joel, seeing her in the distance, the sea is much lower down and the sand isn't wet. (00:03:05)
Continuity mistake: When Joel hides Clementine in a memory of him when he was four, he is hiding under the table crying and Clementine is trying to comfort him. The shot seen through Joel's eyes, while she bends over to comfort him, she tosses her cigarette away. The shot changes to behind Clementine, and you see her reach up to the table to put out the cigarette she already tossed away. (01:02:30)
Continuity mistake: When Clementine walks up to Joel on the train, he is looking directly at her. In the next shot from her left, he is bashfully looking down. (00:05:10)
Continuity mistake: When Joel is suffocating Clementine when they are little kids, initially, the house has a window that is offset to the right, but later on, when the movie returns and the house is boarded up, the window is centered and the house looks different. (01:13:05 - 01:13:45)
Answer: If I remember correctly, it's Mr. Blue Sky by The Electric Light Orchestra (ELO).
umathegreatstationarybear