I remember those. They were also known as “flying saucer” records, after the first one popularized by Dickie Goodman. My favorite one was “Watergrate”.
Edit: Never mind. I was thinking of the wrong song.
Ain’t Gonna Bump No More (With No Big Fat Woman) Joe Tex
Along Came Jones Ray Stevens
*Hot Rod Lincoln * Charlie Ryan
They’re Coming to Take Me Away.
*The Marvelous Toy * Peter, Paul and Mary
“Mrs. Brown, You’ve Got A Lovely Daughter”, by Herman’s Hermits. I remember getting thoroughly sick of it when it was #1 for more than nine weeks one summer. Mind you, I was only six or so, but my older siblings listened to it constantly.
Aww, I love all of these! I still hear some of the not-quite-only-novelty ones on the local oldies station in L.A. (KRTH)
Part of the problem is that “oldies” used to mean 50s, 60s, and a few 70s. I feel REALLY old when I hear “my” generation of songs on the oldies station. Like Pat Benatar. and Billy Joel. 
WVLI (linked above) still has the “old” in oldies; they have two hours of The Beatles on Sunday mornings and Sinatra & The Rat Pack for two hours on Sunday nights.
Turning Japanese - The Vapors
I heard on Sirius the other day. on the 80’s channel
Leader of the Pack - The Shangri-Las
Chantilly Lace - The Big Bopper
Bread and Butter - The Newbeats (one of my personal favorites)
I’m only 26, so all of my knowledge of old goofy songs comes from a cassette I had when I was a kid called “Goofy Greats”
That’s my favorite song! (I have the 45.) Cha-cha-cha!
(* Yes, that’s the original, at least the Top-40 original, maybe not the down-home bluesman with a broken cigar-box homemade geetar original.)
And its companion Bertha Butt Boogie.
I am not a fan of The Three Stooges. I don’t think the guy doing the Curly impression is good. That said, I am the proud and happy owner of a 45-rpm vinyl recording–which I bought new!–of The Curly Shuffle by “Jump N The Saddle”.
“Sky Pilot” by Eric Burdon and the Animals
That’s your idea of a silly song 
If by silly, we mean dislike, then no.
Whispering Grass, Windsor Davies and Don Estelle. If you don’t know the 70s UK TV show It Ain’t Half Hot Mum you may be confused.
I actually don’t think of Who Put the Bomp as a novelty song, and generally it isn’t treated as one on oldies radio. I think maybe Mr. Bassman is treated as novelty. These examples make the dividing line unclear.
But the OP does specify ‘fun’ songs, which is not necessarily the same thing, though songs considered novelty get less play.
Shoop Shoop Diddy Wop Cumma Cumma Wang Dang - Monte Video
Right Said Fred - Bernard Cribbins
Space Invaders - Player One
Funky Gibbon - The Goodies
Buckner & Garcia, IIRC.
Has anyone mentioned “General Hospi-tale” by the Afternoon Delights yet?