Ok, here's what she does in the Season 34 Primere, the karaoke episode. She doesn't actually sing a karaoke song. Every now and then, between songs, she would walk up to the microphone and sing when she wasn't supposed to. She sang 3 different songs that had the word "Moo" in them. And Alan would step in right away, take the microphone off the stand and say, "Thank you, Gladys."
In the episode where Elmo falls in love with Gina, are you sure that was Gladys?