Continuity mistake: As Shirley Jones is singing "In love with Me" at one point, she's sitting down, and is slouching back. The shot changes to the side, and she's sitting up straight, and is much higher against the counter than she was. Her hands have changed position as well, she's no longer holding onto the cup she had in her left hand. (01:19:55)
Continuity mistake: Mayor Shinn thanks the crowd for attending the Fourth of July exercises indoors at the gymnasium due to "the weather being so chancy" (meaning a good probability of rain). Mayor Shinn adds that the July 4th fireworks display will be held that evening unless the rain prohibits it. However, the year is 1912, and no advanced weather-tracking technology exists; so, they have no way of predicting the weather except by direct observation. Therefore, the mayor must be basing his cautionary weather statements on direct observation of overcast and stormy conditions. But, when Professor Harold Hill works the crowd into an excited frenzy that bursts out of the gymnasium and into the streets, it is revealed that the sky is perfectly clear and blue, with the sun shining brightly. There is no indication of inclement weather whatsoever. In fact, it remains clear and never rains in River City throughout the course of the entire movie.