Good Night, Dear Heart - November 9, 1957 - S2-E17
Revealing mistake: When Sam is putting the locket on her in the coffin, Hilla's "corpse" is breathing. (00:39:00)
Pool Hall Blues - September 4, 1954 - S2-E18
Revealing mistake: After Al comes back when the power is turned on, Sam wins a game, breaks for the next and then we see a wide angle shot and the noise is made when Al shows Sam where to shoot. But the beam doesn't appear, even though the shot is seen by viewers.
Sea Bride - June 3, 1954 - S2-E21
Revealing mistake: At the beginning of the show they go to great pains to show (on more than one occasion) the "reflection" of Sam in the mirror. However, when the girl shoves him against it, kissing and slapping Sam, it is his real reflection.
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.
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.
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.
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)
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 Song for the Soul - April 7, 1963 - S4-E15
Revealing mistake: When Al jumps into the shot out of nowhere on the steps after the record producer leaves, Sam and the two girls abruptly move, giving away the technique to achieve Al showing up. The transition is usually smoother, this one happens to be rough and noticeable.
Leaping of the Shrew - September 27, 1956 - S5-E3
Revealing mistake: The background while Scott is on the raft with Brooke Shields is blatantly a studio wall. The clouds have streaks on them with dark colors and the seam where the ocean meets the wall can be spotted. The film "The Truman Show" used the same technique.
Deliver Us From Evil - March 19, 1966 - S5-E7
Revealing mistake: When Ilia is spurned by Sam, she sits on the bed and rips her clothes. When she goes to scratch her face, you can see the fake blood on her hand when she goes to smear it on her face.
A Tale of Two Sweeties - February 25, 1958 - S5-E12
Revealing mistake: Sam's reflection in the mirror near the beginning of the episode drops his handkerchief too quickly after wiping his mouth, making things not match up.
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