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