Other mistake: In Gene Kelly's 'Singing in the Rain' sequence, the taps he makes do not match up with the actual movements of his feet - they are much more complicated than the steps he makes. Since one can't tap dance in wet loafers, Kelly had to do the dance in the studio, and the taps were added into the picture. He apparently got a little overzealous.
Continuity mistake: In the scene where Bing Crosby is talking to Ethel Barrymore at a picnic day for the girls of he very snobbish school, he makes a big deal out of taking out a cigarette and lighting it, but in the next shot - no cigarette.
Revealing mistake: When the Prince and Princess are catapulted over the wall of the Giant's castle, you can see their shadows on the "sky" above the wall; the sky is a studio wall painted blue.
Factual error: In Danny Kaye's song about "The Emperor's New Clothes" he persistently and repeatedly uses the words "the king is in the altogether." Hans Christian Andersen lived from 1805 to 1875, but the expression "in the altogether", meaning naked, was invented and popularised by George Du Maurier in his novel "Trilby" which was not published until 1894.