Private Dancer - October 6, 1979 - S3-E14
Revealing mistake: When Sam is studying the sign language textbook and Al pops in, Sam's right hand makes a noticeable jump, though it's supposed to look like a continuous take. (00:20:10)
Piano Man - November 10, 1985 - S3-E15
Revealing mistake: When Sam is driving the truck through the fog/smoke, Al tells Gooshie to center him on Lorraine but Al doesn't jump out. The shot cuts to a front view of the fog. The road noticeably jumps forward because there is a dissolve transition in this shot, even thought its the same angle. It's likely it was done to lengthen the shot.
A Little Miracle - December 24, 1962 - S3-E10
Revealing mistake: The fractal of the star stops at a point just below the top of the frame, when in reality it should keep going. This gives away that the star is superimposed over the shot. Oddly enough, when it starts snowing, the effect looks normal.
A Little Miracle - December 24, 1962 - S3-E10
Revealing mistake: The video image over the building that Al shows Michael Blake is placed incorrectly. The edge of the effect is blatantly visible. This effect is corrected later when Blake asks to see the plaza again and the name is changed.
Leap of Faith - August 19, 1963 - S3-E3
Revealing mistake: The newspaper Father John holds up has the headline "Youth Hit By Train", but the article text under it talks about Diane Feinstein's election race against John Van de Kamp for California governor in 1990, the year the episode was filmed. Other article texts do not match either. The text under the Bangladesh Flood article talks about Germany's reunification, which also happened in 1990, after the Soviet collapse. The paper is a copy of the Los Angeles Times with the titles and date changed. (00:05:10)
Miss Deep South - June 7, 1958 - S3-E6
Revealing mistake: Connie and Sam are talking in their room, while Sam is looking at his female self in the mirror. As Connie walks back into the shot, her shadow is cast on the wall behind the mirror. This indicates she came from the opposite direction that is possible in the reflection had this not been the mirror trick. In reality, she would have been coming from behind Sam, not from in front and wouldn't have cast a shadow on the wall with the mirror.
Future Boy - October 6, 1957 - S3-E13
Revealing mistake: Just after Mrs. Stiner delivers the line "My car's out front" after her father's hearing, everyone except Al rushes away from the broken window. Sam brushes up against Al's shirt sleeve, causing it to move. Al, being a hologram, shouldn't have been touched. (00:41:50)
Runaway - July 4, 1964 - S3-E11
Revealing mistake: The family crosses into Colorado from Wyoming and calls out the state names. Wyoming doesn't have Joshua trees, nor anything close to the desert we see. This episode was filmed in Southern California.
Chosen answer: Per the Quantum leap page at http://www.scifi.com/quantum/episodes/season5.html. 8 August 1953: An enigmatic leap lands Sam in a Pennsylvania tavern, as his own grown self on the day of his birth. As Al and Gushie work frantically to locate him, Sam befriends a wise bartender (popular character actor McGill, who'd appeared in a different role in the very first "leap") and a group of coal miners. As a host of familiar-looking faces pass through the bar - with different identities than Sam remembers - Sam ponders his life of leaping with Al the bartender, who tells Sam he controls his own destiny. Pressed for more, Al the bartender simply shrugs and says, "Sometimes, 'that's the way it is' is the best explanation." Sam realizes he must right at least one more wrong before he can go home, and leaps back to tell Al Calvavicci's wife Beth (from "M.I.A.") to wait for Al, who will survive Vietnam and come home to her. The closing title cards state that Beth and Al have four daughters and will shortly celebrate their 39th wedding anniversary ... and that Sam Beckett never returned home.
Boobra