The two iconic comic props (as opposed to props often used by famous comics, with which the rest of the thread has concerned itself) are, in order:
The slapstick; and,
The custard pie.
The ladder is another early and iconic prop, but, while it outstrips everything mentioned so far, it comes in third to the above.
I LOVED him! Damned if I can ever remember his name! No one ever knows who I’m talking about.
Thank you WotNot.
Er…Not actually.
You’re probably confusing Pee-Wee Herman with the character Pee Wee from the Porky’s movies.
Pee-Wee Herman is a character played by Paul Reubens in a stage show, two movies (neither of which involved bawdy hijinx), and TV series.
To Groucho’s mustache and glasses I add his clawhammer coat.
To Harpo’s horn I add his broken top hat and bedraggled raincoat (not to mention his harp).
Chico wore a very identifiable pointy hat.
A carrot.
Nyaaaaaaaahhh . . . <crunch crunch crunch>
That might be partly true - as I say, I haven’t exactly immersed myself in the genre.
OK, I think you’re right. I think I’ve just lumped them all together on the basis of certain visual elements, which I’m finding hard to quantify, but there’s a certain ‘feel’ to some American comedy films - something to do with the visual mix of colours, the pace of the action, the mannerisms and gestures of the actors.
Ewww.
You are a sick, sick man. 
Well, it would be a very… unique prop.
Henry Blake’s U of I sweater?
Radar’s Teddy bear?
Les Nessman’s Silver Sow award?
Wait a sec! That’s a comedy prop for Carmen Miranda? I thought that she played things straight in her movies and shows. I grant that everyone else using that prop were milking for the comedy value, but as a spoof of her style. So, while the hat is iconic, and a comedic prop, I can’t think of anyone who used it as a comedy prop who wasn’t lampooning the image that Carmen Miranda established.
(Mind you, I’m a bit young to have actually seen her movies, so I may well be wrong with how they were played…)
A piano. Like what always falls and crashes to the pavement in early comedies like Laurel and Hardy.
A top hat, like W.C. Fields’, which rolls down from the head to the hand and back again.
An anvil, from just about any Warner Brothers cartoon.
The growing lump on somebody’s head, usually because of the anvil.
Explosives like the TNT stick and the black ball with fuse.
Pies. The throwing kind.
A toilet plunger.
Underwear: either when somebody gets their clothes ripped off, or when it’s pulled out of somebody’s pants.
Doggie doo.
Fake vomit.
Teeth: the chattering kind, dentures, buck teeth, or missing teeth.
A crate from ACME.
Shoe phone. Complete with rotary dial!
A lime green Mini (car that is).
:: puts on Grammar Police hat ::
Couldn’t be “very.” Something is either unique or it isn’t.
Wow, that’s alot to remember.
A 16-ton weight
An invisible golf club (think about it)
Blue note cards
A guitar
also:
a whisky jug with XXX on the front
a black eye or two
lipstick prints
really fat people
AND A RUBBER CHICKEN!
I bet if I said, “flapping dickie” most people would only think of one character.
“me-a frap-a dickie rong time”
Victor Borge’s piano bench with built-in seatbelts.