On the plane trip to Philadelphia when drinks are served they are in a clear plastic glass not manufactured until the mid-1970s. In fact airlines of the 1950s used the real thing, usually etched with their logo. Even the dinnerware was real then.
When Ritchie and Donna are riding in Donna's red '49 Ford convertible, on the radio is "This I Swear" by The Skyliners. The song was not released until May 1959, three months after Ritchie Valens' death.
When Ritchie flies to Philadelphia he is seen flying in a DC-3 which wasn't the type of plane that was making transcontinental flights in the late 50's. The interior mock up is enormous with what looks like a cathedral ceiling, huge seats and a wide aisle. TWA was flying SuperConstellations, United DC-7s, cheaper airlines DC6s. The Boeing 707's would come into service the next year (1958).
The bassist of the fictional backup band "The Silhouettes" is playing a Harmony H-22 bass guitar. Harmony didn't make bass guitars until 1962. The most common electric basses made in the 1950s were by Fender, Gibson and Kay.
When Richie and his brother Bob get into a shoving match, Bob pulls on Richie's sweater and yanks his tie out of kilter. A few seconds later, Richie's tie is tucked in and his sweater is smoothed down, but not enough time has elapsed for him to have done this to himself.
When Connie goes to tell Ritchie she was leaving for work, she opens the door and steps down into the basement to stand on the top step. When the camera angle changes to where Ritchie was in the basement, her foot isn't on the top step.
When Ritchie is showing his mother the new house, the shadows are halfway up the walls. When Bob pulls into the driveway with his two sisters, the shadows are totally gone. And when Ritchie opens the door into the house the shadows are there again.
When Bob takes Ritchie to the brothel, Ritchie chooses to relish the music rather than the women. Bob is left with the women and as Ritchie approaches the stage, you can see Bob in the background putting his arms around a woman. In the next shot, Bob is just approaching the same woman.
When the DJ announces the sad news of the plane crash, he plays the song "Sleepwalk" by Santo & Johnny as a tribute to the fallen musicians. The crash occurred on 3 February 1959 and the song "Sleep Walk" was not released until July 1959.
While the scene is true about the coin toss between Ritchie and a member of Buddy's band, Ritchie Valens was not sick as seen in the movie. The Big Bopper was. He begged Holly's band mate (and future country music star) Waylon Jennings for the chance to ride on the plane because he was coming down with the flu. The coin toss took place between Ritchie Valens and Tommy Allsup.
In one of the final scenes, you see Buddy Holly and his band performing "Crying, Waiting, Hoping". In fact this song was never recorded in the studio or performed with his band during Buddy Holly's lifetime. The song was recorded privately by Buddy on his own with his guitar as an idea, on a home tape recorder. But not until after his death was it made into studio release, by various musicians recording overdubs. It is extremely unlikely Holly would've played this song that night.
The morning after Ritchie and Bob played in the Cowboy Palace, there is a 1957 calendar shown hanging in the kitchen but the days are incorrect in correspondence to the dates. The calendar shows October 1 to be on a Wednesday when, in fact, it was on a Tuesday.
During the party for Ritchie, his mother takes a picture with a Kodak Brownie camera. She holds the camera up to her eye which seems normal, however, that style of camera has a viewer which requires the photographer to look down through a prism and not through an eye piece. Normally the camera was held at belly level and while the user looked down through a prism which reflected what was in front of them - it would not be held up to their eye as is depicted.
At the Pizza Party featuring a personal appearance by Ritchie, Bob Keane is handing out copies of Ritchie's first album. This is erroneous on two accounts: (1) The album wasn't released until March 1959 (one month after Ritchie's death), and (2) as the actual album features "Donna" and "La Bamba", Ritchie had not yet recorded either song, nor had he even written "Donna" yet, at that point in the movie.