Continuity mistake: When the woman is about to be sucked by the void and fiercely grabs Boris' wrist, the wide and close-up angles show her using her left arm, however in several close-ups it's the right arm shown, as noted by position of her thumb, style of the bracelet (from grey rubber to black with diamonds), and a ring appearing/ disappearing. (00:05:10)
Continuity mistake: At the Chinese restaurant, when K and J are eating, J's white napkin keeps moving around between shots, as do some of the saucers and cups on the table. (00:13:45)
Continuity mistake: When J drags Warhol backstage, the people they pass-by have no continuity between the front and back shots. Note the tall transvestite with bulky hair they bump into first, who, a shot later, appears several meters ahead. (00:55:45)
Continuity mistake: During the scene where Agent J and K are driving around Midtown Manhattan, the wide shot shows them driving past the red TKTS staircase in the middle of Times Square. Then there is a shot of J inside the car a few moments later, and they pass by the red TKTS staircase again. (00:09:50)
Continuity mistake: When J is being eaten by the fish, his trousers are clean. Right after it growls, a close-up shows his trousers covered in mustard, despite him not shooting it until several seconds later. (00:16:25)
Continuity mistake: When J peeks out The Factory window the broken glass around the frame change amount and shape between shots. (01:01:25)
Continuity mistake: When K slaps the Chinese man with the alien crab his hair gets messy. When the shot changes it's back to brushed. (00:12:45)
Continuity mistake: At the diner, J says to the waitress K has apple pie. K is then given some apple pie. In one shot, the waitress is in the middle of giving K the plate with pie and the plate has not reached the table yet. But in the immediate shot afterwards, the pie has already reached the table. (01:03:45)
Continuity mistake: When they are leaving the first crime scene (the crashed UFO) in their Ford Taurus they are shown talking with headrests in the vehicle. They arrive at the headquarters for Zed's eulogy. After the eulogy they are back in the Ford Taurus and the headrests are gone. (00:07:40 - 00:09:50)
Continuity mistake: When Boris is about to french kiss the woman, she opens her mouth wide and approaches him; then from the opposite angle the mouth is closed and she is walking towards him again. (00:02:30)
Continuity mistake: When J and K fall off the roof holding onto a door, they land opposite to each other. A frame later, from a different angle, they're laying next to each other. (00:17:55)
Continuity mistake: When J steals the blue car keys, a nerdy looking guy with an orange and white striped shirt and glasses is walking towards the hotel. After J drives zooming through New York, the same guy is seen crossing the street walking the opposite way. (00:39:15)
Continuity mistake: When J exits the Flatiron building in 1960 a woman walks right behind him, then the shot cuts to his POV of a group of hippies, then back to the previous shot of J and the woman obviously continues her walk. She should've been several meters ahead already. (00:38:30)
Continuity mistake: After K is falling with J from the rooftop, big chunks of concrete start to fall, yet disappear in some shots. When they land, the ground is clean, yet in one of the close-ups of K a chunk magically reappears next to his head. (00:17:50)
Continuity mistake: In the Headquarters in the altered time line, when J is crying out that he and K spent 14 years of stakeouts, cowboy music, etc. A woman in the back swaps from standing by a screen to walking 20 feet to the right, between shots. (00:28:40)
Continuity mistake: As Griffin is telling J that K has to go to the Apollo launch, the shot from his perspective shows J holding his nose from where K punched him, his hand being over his nose. The shot cuts to show Griffin's face and suddenly J is wiping the underside of his nose. (01:16:35)
Continuity mistake: When Jeff is talking to J on the Flatiron, he is looking at J except for a brief shot where strangely he is suddenly facing left without having turned his head. Then a frame later, from another shot, he is back facing J. (00:33:50)
Continuity mistake: When K tells J he's suspended after asking about Boris, they get into a brief confrontation about it. As K is storming away to take the car, leaving J back at Wu's without a ride, J's suit is still dirty, wet, and nasty from his fight with the alien fish. Moments later, J is seen going to MiB headquarters and now his suit is clean and in perfect order. (00:19:45)
Continuity mistake: After Boris has blasted a hole in the steel wall of Lunar-Max, Boris grabs onto his girlfriend's arm who is getting sucked into the vacuum of space. The first shot shows his hand wrapped around her wrist. A few shots after, his hand is further up her arm. (00:05:10)
Continuity mistake: Boris' girlfriend is first shown to have a tattoo with "Boris" over her back. A few minutes later, after Boris says "I'm a bit of a romantic", the Lunar-Max guards walks over to the cake in Boris' girlfriend's hands and the tattoo on her back is shorter. (00:00:35 - 00:03:00)
Chosen answer: The name of the song is "Love is Strange," recorded by Mickey and Sylvia, and released November 1956. Pitbull sampled it in his song "Back in Time."
Chanteuse66