The Lalor family watches Duran Duran's video for "Rio" - Duran Duran: Rio - on Top of the Pops. Brendan claims it could go either way as to whether or not they succeed; however, the song was released in 1982 and the film takes place in 1985, by which time Duran Duran was already an extremely successful band and a household name.
During the music video they film in the school, Cosmo mentions the movie Back to the Future; however, Sing Street is based in 1985 and Back to the Future was not released until 20 December 1985 in Ireland. With the time frame of this movie and no indication of him seeing Back to the Future during this time, it makes no sense for him to be aware of what it is.
While filming the "Riddle of the Model" music video, Eamon knocks one of the drummer's cymbals with his guitar while doing a spin. The cymbal vanishes in the next shot when Conor sings at the microphone.
As Conor leaves his house, the day after learning that Raphina has left for London, he's wearing the light brown shoes that he painted black to comply with Brother Baxter's order. In the next scene, they're black again.
When the Lawlor family watches the "Rio" music video, Ann asks Brendan "Who is that guitarist" (John Taylor). Brendan responds he is cute, and teases her, but says John Taylor is a bassist, not a guitarist. The video does not show either instrument, and since Ann had never heard of Duran Duran, it is impossible for her to know that he is a guitarist, or a bassist, or whatever.
The goof item below may give away important plot points.
The film ends with Conor and Raphina leaving harbor to cross the Irish Sea in a small boat. Brendan releases its aft mooring rope, but another rope connecting the prow to the jetty and a second boat on the other side simply disappears.