Not a singing moment per se, but this thread is reminding me of one of my favorite musical "Best Show" moments ever -- where Tom repeatedly played the alt mix of "'Til I Die" while multiple callers were speaking over each other. If I remember correctly, Tom was threatening to end it all by eating a roll of paper towels, thereby preventing him from singing.
Also, can't get me enough of his renditions of "Kids" from Bye Bye Birdie.