Personally, I was okay with the Row, Row, Row Your Boat, but I absolutely cringed when Picard and Worf were singing "A British Tar" from HMS Pinafore, complete with LCARS displaying the karaoke bouncing ball. Ugh.
See, I don't see that as a low point. Data's character still acts, even with emotions. It's within his character and actually moves Data's character along. The last time we saw him acting was in the series, and if you remember Devil's Due, he explains why he acts, as a part of his quest to understand humanity. Picard was often his audience, as he probably was for this play. Picard's dialogue before they start singing shows the mind at work in Picard:
"He can fly a ship. He can anticipate tactical strategies. Clearly, his brain is functioning. We've seen how he responds to threats, I wonder how he would respond to...Mr. Worf, do you know Gilbert and Sullivan?"
And this is where I get on my high horse about "having them act as scientists, explorers, rather than just holding a phaser and shooting at someone." (Deducing, having an hypothesis from known data. Testing said hypothesis. Scientific method...yum.)
If you consider the song they are singing, Data is acting with duty and honor, as a British Tar should do. It fits the scene. It is no more out-of-place than Chang quoting Shakespeare, Khan doing Melville, or Kirk doing Dickens.