I remember singing the Yumblie, Bart Simpson, et al version.
And also
*Randolph the six-gun shooter
Had a very shiny gun
And if you ever saw it
You would turn around and run
All of the other cowboys
Used to laugh and call him names
They never let poor Randolph
Join in any poker games
Then one foggy Saturday
The Sheriff came to say
“Randolph with your gun so bright,
Won’t you shoot my wife tonight?”
…*
When I was in junior high, I heard a similar one about Ruby, a hooker. I shouldn’t say what she had that was shiny, but seeing it would make the other hookers grunt.
I learned as Randolph the lame-legged cowboy. The final verse is
Then how the cowboys cheered him
As they danced around in glee
Randolph the lame-legged cowboy
You’ll go down in in-fa-my!
I’m Chiquita Banana
And I come to say:
Get rid of your teacher
The easy way –
Put a banana peel on the floor
And watch her go sliding
Out the door!
Jesus Christ, Superstar
Ridin’ down the highway on a Yamaha
Cops are there, I don’t care
I’ve got bullet-proof underwear!
Ahhh, kids these days…err, I mean, those days. Back then, that is.
I learned this one as:
*Joy to the world, the teacher’s dead
We barbequed her head!
What happened to her body?
We flushed it down the potty!
And all the children sing … *
Jingle Bells
Briston smells
Fifteen miles away
Blows his nose
In Cheerios
And eats them every day.
Briston used to keep the rhythm. I have no idea how he smells, although it should be said I live more than 15 miles away from him. I’m sure he smells just fine. At least he did once, for 20 minutes, about 10 months ago.
I’ve always heard this on as:
Mine eyes have seen the glory of the burning of the school.
We have tortured all the teachers, we have broken every rule.
We have bar-b-qued the principal, broken the PTA,
Us kids keep marching on.
'Round and 'round it went
'Round and 'round it went
'Rou-ound and 'round and 'round it went.
My elementary schools version of Batman/Jingle Bells was
Other Christmas favorites:
Hark the Harelipped Angels sing,
"Gwowy tu da newbun ting
and
Silent Night, Holy Night,
me and my brother had a big fight
knocked the Christmas tree onto the floor
the police had to break in the door
Jesus and Mary were mad!
But oh what good turkey we had!
My first grade teacher taught us the “Rubber Cigar” song and got into BIG trouble for it (Christian school and all).
We had this ditty:
Joy to the world, the school burnt down,
and all the teachers too!
The principal is dead,
we shot him in the head,
the secretary too,
we flushed her down the loo
Rejoice, rejoice, the school burned down
And in answer to the OP, our version was:
Jingle Bells, Batman smells
Robin flew away
The Batmobile lost its wheels
and found them in Bombay
Hey!
The version my mother taught me:
Jingle bells, Batman smells
Robin laid an egg
The Batmobile lost a wheel
And the comissioner’s eating hay
I also recall a version wholly unrelated to the Dark Knight repeated ad nauseum by some of my classmates whose families held decidedly outmoded views regarding cultural diversity, which included the charming couplet
Pulled the trigger
Shot a N****r
When a visiting friend recited this version at my house, my mom promptly disinvited him for several weeks.
OK, then I’ll disagree with the guy in post #3.
And Robert Goulet in the treehouse casino:
Goulet: Are you from the casino?
Bart: I'm from *a* casino.
Goulet: Good enough, let's go.
Jingle Bells
Batman smells
Robin flew away
Wonder Woman lost her bra
Flying TAA.
TAA is a now defunct Australian Airline (Trans Australia Airlines - I think… 
)
Batman: The Animated Series had a Christmas episode that began with Joker singing the version of the song that end in “And Joker got away”.
As a kid, I learned the versions with him taking ballet, laying an egg, and breaking a leg. I think all the other lines were constant.
Another such song that I loved was from Woody Guthrie’s “This Land is Your Land”
This land is my land
This land ain’t your land
I’ve got a shotgun
You don’t got one
I’ll blow your head off
If you don’t get off
This land was made for me, not you
We always sung the last line as “this land is private property.”
Another one I remember:
*Row, row, row, your boat, gently down the stream.
Throw your teacher overboard and listen to her scream.
Five days later, floatin’ in her underwear
Wishing she had another pair.
Five days later, polar bear ate her
…and that’s how the polar bar died. *
The second two verses didn’t follow the * Row, Row * melody too well, but we sung 'em anyway.
But the thing of it I , I knew this rhyme as well. So how did things like this travel from where Hal is (which I assume by his location is in the US) to where I was, which was on top of an ant hill in the mulga of Western Queensland in 197…2? without the miracle of, say, the internet? What a wonderful world!
mm
Man, nuny’all sing 'em right.
Jingle Bells,
Batman smells,
Robin laid an egg.
Blows his nose in Cheerios
And eats it every day.
Also:
Deck the halls with poison ivy
fa la la la la, la la la la
Light a match and throw it i-in
fa la la la la, la la la la
Watch the school house burn to ashes
fa la la, la la la, la la la
Aren’t you glad you played with matches
fa la la la la, la la, la, la.
Also:
*We three kings of Orient are
Smoking on a big black cigar
It was loaded it explo-o-ded
POOF and there’s only two.
We two kings of Orient are
Smoking on a big black cigar
It was loaded it explo-o-ded
POOF and there’s only one.
I one king of Orient am
Smoking on a one gre-ate big ham
It was loaded it explo-o-ded
POOF*
Also (sung to Old Smokey):
On top of the schoolhouse
All covered with blood
I shot my poor teacher
With a .44 slug.
I went to her funeral
I went to her grave
Most people threw flowers
But I threw grenades.
Also (and I’m just throwing it in for auld lang syne):
*My Bonnie has tuberculosis
My Bonnie has only one lung
She spits up a bloody solution
And sometimes she chews it like gum.
(sung in harmony with traditional chorus)
Bring back my Bonnie
Don’t try to be funny
Oh bring back my Bonnie to me, to me!
Bring back my Bonnie
Don’t try to be funny
O bring back (go up here!) my (higher!)BON (ok, back down) nie to me*.
Man, two people now have done the “rubber cigar” version of Three Kings, but no one apparently knows how to *properly *finish the song:
♪♫ Silent night… ♪♫
Deck the halls with gasoline
fa la la la la, la la la la
Strike a match and watch it gleam
fa la la la la, la la la la