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)
Glitter Rock - April 12, 1974 - S3-E17
Other mistake: During the album signing at the mall, a woman exposes her breasts to Sam. We see her from the front with Sam's hat covering her breasts. His hat moves just enough that you can tell she is wearing something covering her breasts.
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