Uhura was flirting with Spock, and he was in the mood to play along (both literally and in the sense of camaraderie - or maybe it was just that Leonard Nimoy had trouble keeping a straight face during the song?). I maintain that they had a properly respectful professional relationship, and a respectful friendship (since they share an interest in music, that makes sense). At no time would I expect to see Original Spock and Original Uhura kissing anywhere, never mind in public, or while on duty.
That's your interpretation. And it's a valid one -- but so is the interpretation of Orci and Kurtzman that it's perfectly possible that the two were in a relationship
My point was, that relationship wasn't some crazy thing they threw in there out of nowhere -- there's evidence in TOS to support the notion, primarily in "Charlie X" and "The Man Trap."
So it's valid to say that every time two or more TOS characters treated each other with respect, professionalism, or camaraderie, they were in a more intimate relationship? That's ridiculous.
This conversation reminds me of RL conversations with people who insist there is no way that adult men and women can be friends without sex being involved. Sometimes a friend is just a friend.
EDIT: I call BS on the bolded part. If they honestly think Original Spock and Uhura were in a relationship on TOS, they must never have seen "Amok Time." Spock was so embarrassed about the topic of sex, he could barely bring himself to tell Kirk about it, never mind anyone else (don't bring up Droxine; that scene never made any sense at all). And there was no flirting or anything else between Spock and Uhura after
"Amok Time" so I don't see any evidence whatsoever that they went past casual friends.