They also have both immortalized a particular street story that must have been from around 1986 (I was three) in which Elmo is going around showing off his brand new sneakers to everybody, saying over and over "Elmo has new shoes!"
Something about that line and the way Elmo said it struck both my parents very funny, and they've kept it alive ever since. They used to just randomly say it when someone got new shoes .
The only part of the story I remember is Elmo bugging Maria about the shoes while she's busy with something, and eventually he either trips and falls down out of camera range, or she somehow CAUSES him to fall and smiles in self-satisfaction. (Yeah, I know that second one is unlikely, but that's how I interpreted it as a 3-year-old and that's how I remember it...)
Oh yeah---and my younger sister was always crazy about "Number 9 Cutie" (the Bud Luckey cartoon). We used to sing it.
Mom liked to imitate the way I sang "ABC-DEF-GHI"; I must have really mangled Big Bird's pronunciation back then, because Mom's version ended with "ka-super-fizz".
Dad used to sing the "Number Song" intro whenever he fiddled with a calculator. He'd press a digit repeatedly to show off and sing, for instance: "3-3-3-3...let's sing a song of 3. How many is 3?"
My younger brother used to torment me with unscripted re-creations of the "P for Pillow" cartoon--in which I got hit mercilessly with as many big pillows as he could grab. (And yes, my brother quoted "P...Pillow!" in that cartoon boy's voice before he whacked me!)
I remember asking my mom as a teen what she remembered/liked, and she basically enjoyed the skits/characters that she saw me really get engaged by as a toddler. She paid more attention to the characters than individual skits, but she thought Grover was adorable (natch), and got a big kick out of Oscar and Cookie Monster, as well as Bert & Ernie. The ONE sketch she remembered almost perfectly was "Count to 20 with Herry and John-John"... She said that was one of the cutest things she's ever seen.
My first girlfriend (7th grade) and I got together due to a moment in gym class where I imitated the yip-yips, which she had forgotten about until then... she got this BIG smile on her face, saying "...OH YEAH! Yyipyipyipyipyipyip, uh-huh, uh-huh... te-le-phone! BRRRRRing..." The rest of the day, whenever we saw each other we'd start yipping away and laughing hysterically.
Most of my family doesn't watch Sesame Street, but I still do (when I get the time to) and I just recently saw a re-airing of Bip Bippadotta singing "air"! BTW, there's a full video of him singing "That's opposite" on Youtube! Bip Bippadotta's skits are hilarious (IMHO)
I remember as a kid my dad liked Frazzle, (the song as well as the character himself) and would often ask me to sing the song, which I still know by heart all these years later.
My brothers and I always enjoyed "What Do You Do With A Fruit", we still sing that one even now. Funny thing is that my daughter who is now 18 months old watches SS, but is all about Ernie not Elmo like most of today's kids are. Her favorite bit with Ernie is "Imagine That".