The real golden age of the novelty song was the 1920s:
Crazy Words, Crazy Tune
Barney Google
Yes! We Have No Bananas!
Horses, Horses, Horses
I’m Just Wild About Animal Crackers
She Knows Her Onions
I Married the Bootlegger’s Daughter
Ain’t Gonna Rain No Mo’
That’s a Lot of Bunk
etc., etc. . . .Warren Zevon’s * Mr. Bad Example. * Despite having heard this song literally hundreds of times, its sardonic humor still puts a grin on my face when I hear it.
“Dead Puppies” is credited to Ogden Edsl according to Dr. Demento.
that would be The Kingston Trio for “With Her Head…”. also known fondly for “Zombie Jamboree”
“An Irish Ballad”
“One day when she had nothing to do
Rickety tickety tin
One day when she had nothing to do
She cut her baby brother in two
And served him up as an Irish stew…
And invited the neighbors in…”
always loved “The Hunting Song”, and who could forget “The Wild West is Where I Wanna Be”? so many goodies…
i seem to recall he still infests the airwaves (though i’ve never actually heard his program). at any rate, as long as he keeps putting out his “Best Of” anniversary collections, i’m a happy girl.
David Allen Coe’s “You Never Even Call Me by My Name”:
I was drunk when my Mama got out of prison,
And I went to pick her up in the rain;
But before I could get to the station in a pickup truck
She got run over by a damned old train!
I’ll hang around as long as you will let me;
And I never minded standing in the rain;
You don’t have to call me darling, Darling;
You never even call me,
I wonder why you don’t call me,
Why don’t you ever call me by my name?
Or Dr. Demento’s perennial favorite, “Shaving Cream” by Benny Bell:
I have a sad story to tell you,
It may hurt your feelings a bit;
Last night when I went into my bathroom,
I slipped on a big pile of…
Shaving Cream, be nice and clean,
Shave every day and you’ll always look keen! 
And there’s always this one:
“They’re coming to take me away,
Ha ha, ho, ho, hee, hee,
To the Funny Farm,
Where life is beautiful all the time,
And I’ll be happy to see those nice young men in their clean white coats,
Annnd they’re coming to take me away!”
Chuck Berry - My Ding A Ling
*When I was a little bitty boy
my grandmother bought me a cute little toy
Silver bells hangin’ on a string
she told me it was my ding a ling
My ding a ling, my ding a ling
I want to play with my ding a ling
My ding a ling, my ding a ling
I want to play with my ding a ling
*When I was a kid, that was on an LP I had called “Dumb Ditties” along with
My mother, a Fundamentalist Pentecostal Christian, forbid me to listen to the Chuck Berry Song. So of course I had to do it in secret when I’d take it to my cousin’s house. I could sing along to The Ballad of Irving without missing a note. I can still remember some of it now.
“Boobs A Lot,” by the Fugs, the East-Coast answer to the West-Coast Mothers:
Do you like boobs a lot?
Yes, I like boobs a lot.
Do you wear your jock a lot?
Yes, we wear our jock a lot. (It doesn’t get much better)
And another Martin Mull hit is “The Blacks Are Giving Me The Blues.”
And not really a comedy, but a novelty song: Swamp Witch Hattie
My dad used to sing these songs to my brother and me when we were little. Good times.
Not magnesium, selenium. Magnesium comes much later in the song.
(BTW, CalMeacham, I’d just like to say that you have the BEST sigs.)
And now I would lke to nominate The Carpenters’ Intermission. Beautiful work.
Stations which broadcast Dr. Demento are listed in a FAQ that’s posted on the Usenet newsgroup rec.music.dementia. There are also webcasts from various stations. In addition, the current show is usually posted by a listener on alt.binaries.sounds.mp3.comedy.
My favorite funny song? There’s too many! Here are just a few of my favorites.
"The Borgias are giving an orgy, there’s a Borgia orgy tonight,
And isn’t is sickenin’, we’ve run out of strychnine,
The gravy will have to have ground glass for thickenin’ " (performed by Hermione Gingold)
“There are fairies at the bottom of our garden” (performed by Bea Lillie)
“Carrot Juice Is Murder” (Arrogant Worms)
“She Loves You” recited by Peter Sellers in his Dr. Strangelove voice.
“So Long Mom (I’m Off to Drop the Bomb)” by Tom Lehrer. Someday I’d like to perform this as a vaudeville song, complete with cane and straw hat which I’ll wave as I smile broadly and sing “No need for you to miss a minute of the agonizing holocaust, yeah!”
“Folk Song Army” also by Tom Lehrer. “We’re all against poverty, war and injustice/unlike the rest of you squares.” Great!
“Lizzie Borden” and “The Boston Beguine” from “New Faces of 1952”
“April in Fairbanks”
Someone is posting Dr. Demento shows from 1974 and earlier on absmc. I just heard one performed by Eartha Kitt -
I’ve never heard the Trio’s version of “With Her Head Tucked Underneath Her Arm” (by R.P. Weston and Bert Lee), but I’m sure it was just a near-to-vanishing pale version of the all-time ultimate best version by the legendary Stanley Holloway in the 30s. And anyone who doesn’t agree is a poopyhead. 
May The Bird Of Paradise Fly Up Your Nose by Lonnie Donegan & His Skiffle Group
Pencil Necked Geek by Fred Blassie
Dead Skunk by Loudon Wainwright III
When I was a kid my dad had an album (KTEL, I think) by various artists called Wacky Westerns or something. I used to listen to it over and over. The ones I can recall are:
Does Your Chewing Gum Lose its Flavor on the Bedpost Over Night?
The Tennessee Bird Walk
Little Arrows
The Battle of Camp Kookamunga
You Can’t Rollerskate in a Buffalo Herd
Chug-A-Lug
Running Bear (loves little White Dove)
How to Catch an African Skeeter Alive
My Uncle Used to Love Me But She Died
How Much is that Hound Dog in the Window?
Three Sheets in the Wind
Doo Waka Doo
and finally,
Mr. Custer
There were more, but these are the ones that stuck with me.
“The Ballad of Irving”
“The Ballad of Ben Gay”
There have been some funny songs released in the past few years.
“My Name Is” by Eminem is probably out-and-out one of the funniest rap songs ever, and one of the funniest songs period.
“Hey Ya” by Oukast–that was a great band totally lampooning the Motown, Atlantic and Buddah sounds while doing a song about breakup. All in one song, showing their brillance.
And there is that “Thong Song” by Sisqo, which he will have a hard time living down.
There was a great song covered in a musical number on The Muppet Show. I would love it if anyone could inform me of an original recording of it, I only know it through the Muppets.
I don’t know the exact title but it’s a song about
The Girlfriend of the Whirling Derbish
(she gives him the run-around!)
.
Any of Ray Steven’s comedy songs. A couple have been mentioned already: “Ahab the Arab” and “Along Came Jones”. But there was also “The Streak”. 
To Musicat: I have “Boobs a Lot” by the Holy Modal Rounders, recorded off the Dr. Demento show about 1990.
To Extraneous: “May the Bird of Paradise Fly up Your Nose” was by Lonnie Donegan and His Skiffle Group, from about 1960. (Billy Jones and Ernie Hare had done it years before).
You know Jones and Hare as the Happiness Boys:
“I’m Billy Jones”, “I’m Ernie Hare,”
“We’re the Interwoven pair,
How do you doodle-oodle-oodle-oodle-oo!”
“Now give me a car that starts with P!”
“Don’t kid me, no car starts with P.”
My Granny is a Cripple in Nashville" by Billy Connolly.
Also his version of Tell Laura I Love Her : “Yes, friends, there were pieces of Laura everywhere…I remember one piece was sliding down the Post Office window and an alsatian ran off with it”.
** Ice Wolf **- Along Came Jones was the Coasters,and was probably Lieber and Stoller’s answer to ** Western Movies ** by the Olympics.
From that general 50s period 2 of my favs have always been * Stranded In the Jungle * and * Alley Oop.*
Curiously, 2 different groups covered each of those.
The Elements (warning: Flash animation! Work-safe, though)