She Really Can Sing: Lynne Thigpen as Luna. It's especially poignant if you knew her best as The Chief, who never sang.
Tastes Like Diabetes: The show occasionally veer towards this with things like the "Dear Grandma" song. It never quite seems to get there, though, possibly because it's just too sincere.
Tear Jerker and Harsher in Hindsight: Listen to The Goodbye Song. Then ponder upon the fact that Lynne Thigpen, the VA of Luna, has passed on. More than 10 years ago. After that, wipe those tears off your eyes, listen to Bear's ending speech, and when he ends with a "see you soon", realize in horror that the show has been off the air for 6 years and has not been in production for 11 years.
A Beary Bear Christmas has one song where Tutter sings about what if there is no Big Blue House. A real tearjerker if you're watching in 2015. As an added irony, according to Word of God, the writer of the song originally intended for the song to be mainly performed by Bear and not Tutter.
They Wasted a Perfectly Good Character: There were a number of regular guests that they probably could have developed more or even become regulars, but on the whole they generally stuck with focusing for the most part on the main cast.