There once was a girl named Lenore
Who wanted to become a whore
She met some Marines,
Let them into her jeans
Now she’s full of esprit de corps.
Since today is St Patrick’s Day:
There once was a man from Limerick
Who bemoaned the bad luck of the Mick
A stout and fistfight
are no use for this blight
Against George’s snakes where’s St. Patrick?!
A rancher who never meant harm
had a penis as long as an arm
All the women screamed ‘freak’
But his cows could not speak
Thus I never drink milk from his farm
There once was a queen of Bulgaria
Whose bush had grown hairier and hairier
'Til a prince from Peru
Who came up for a screw
Had to hunt for her cunt with a terrier
there once was a man from Wheeling
Who did a great deal of dealing
He read on the door,
“Don’t spit on the floor”
So he jumped up and spat on the ceiling.
I like:
A preoccupied vegan named Hugh
picked up the wrong sandwich to chew.
He took a big bite
before spitting, in fright,
“OMG, WTF, BBQ!”
Maths, and also philosophy:
*Bishop Berkeley suggested that “God
Must find it exceedingly odd
To see that this tree
Continues to be
When there’s no-one about in the quad.”
“Dear Sir,
your astonishment’s odd,
I am always about in the quad
And that’s why the tree
Continues to be,
Observed by,
yours faithfully,
God.”*
There was a young man from Racine
who invented a fucking machine.
Concave and convex,
it could serve either sex
and anything else in between!
There was a young man from Madras
whose testes were fashioned of brass.
When he clanged them together
they’d play “Stormy Weather”
and lightning shot out of his ass!
A knight named Sir Christopher Palmer
sold his gear to a young Chinese farmer.
When he heard about this,
King Arthur said “Chris,
I hear there’s a Chink in your armor!”
A topical one, because, well, why not?
The Vatican gives us new hope:
White smoke means an incoming pope
Jorge is now Francis
But what are the chances
He’ll turn out to like the odd grope?
And another (well it was topical a month ago!)
A petrified Russian named Boris
Yelled “What’s that? It’s headed right for us!
Looks rather like
A meteor strike
Won’t somebody send for Chuck Norris!”
An original:
There once was a man from Cologne
Whose mouth had a mind of its own
He frets and frets
That he’s got Tourette’s
SHIT DICK FUCK BALLS
A highly-condensed version of the “Seven Ages of Man” speech from “As You Like It” –
Seven ages: first puking and mewling,
Then very pissed-off with his schooling,
Then fucks, and then fights,
Then judging chaps’ rights,
Then sitting in slippers, then drooling.
(Most of my favourites have been listed in various forms.)
A sailor whose name was McFee
Would spoonerize to a degree
Instead of “Weigh anchor!”
He’d shout out “A wanker!”
And he’d say that his name was FcMee.
There once was a lady from Devon
Who was raped in a garden by seven
High Anglican priests
Lascivious beasts!
Of such is the kingdom of heaven…
A flea and a fly in a flue
Were caught, so what could they do?
“Let us fly” said the flea,
Said the fly, “Let us flee”
So they flew through a flaw in the flue.
Very good! 
Attributed, I think, to the late British author Kingsley Amis.
I always like the ones that play on spelling pecularities:
There was a young lassie named Menzies
Who said, “Do ye know what this thenzies?”
Said her aunt wi’a gasp
“My lass, that’s a wasp
And you’re holding the end where the stenzies!”
The very refined Lady Cholmondely
Was both well-to-do and quite colmondely
But she found introductions
Quite major productions
Because of that name spelled so rolmondely.
Here’s one I just made up:
There once was a fellow named St. John,
Who went to Bombay with an t. John
He said, “Tell you what
It’s frightfully hot”"
Said the t. John, “St. John, stop your wht. John!”
It’s been done before, and a lot better, as:
There was a young student of St. Bartholomews
Whose car went by fits and by St. Startholomews
Till a fewllow named St. John
Examined the t. John
And fitted it out with spare St. Partholomews.
(you can probably figure out how St Bartholomews is pronounced…)
Alternate last line: But man, what a bitch to clean!
There once was a man from the sticks
Who loved to write limericks
But he gave up the sport
Because he wrote them too short
There was a young lady from Pittwood
Who had trouble controlling her shit good.
She’d leave small deposits
In corners and closets
Much as a small pup or kit would.
And my favorites from Edward Gorey:
Augustus, for splashing his soup,
Was put for the night on the stoop;
In the morning he’d not
Repented a jot,
And next day he was dead of the croup.
To his club-footed child said Lord Stipple,
As he poured his post-prandial tipple,
“Your mother’s behaviour
Gave pain to our Saviour,
And that’s why He made you a cripple.”