This is definitely a nebulous, hard-to-define genre, but what with the recent sudden suicide of Dead Milkmen bassist Dave Blood [RIP], it got me thinking.
I like a lot of music that isn't real popular. I also like a good 'funny' song. There are, as far as I can tell, relatively few bands/artists who *specialize* in this [humor is harder to do, especially consistently, than making "popular" music.]
Here's who I came up with [also keeping in mind that I am not up with 'current' music, so my selections are somewhat antique in and of themselves.]

My personal favorites:

Dead Milkmen [I know a couple of the guys had side projects which would fit here too]
Mojo Nixon
Weird Al Yankovic
King Missile
Dick Nixons
Left Wing Fascists
You could *maybe* put SOD/MOD here, based on their subject matter.

Others I know of or have heard of:

Julie Brown
Ween
They Might Be Giants
From what I know of Tenacious D, they might fit here.
Lawnmower Deth

And that's all I can really think of, without browsing my cd/cassette collection or surfing Google.

Anyone else have any suggestions of [English-speaking] bands/artists in this genre, new or old?

Butterfly Joe would probably fit, but I don't think Burn Witch Burn would fit (but still BWB is absolutely brilliant check them out!). Off the top of my head those are the only sideprojects I can think of, though I know there are more.

Butterfly Joe is the one I was thinking of, I hadn't listened to BWB, I'll look for it.

"Don't Call Me Dude"

Green Jellÿ(ö)'s "Cereal Killer Soundtrack" was pretty great. And there's always GWAR.

the Dead Kennedys always crack me up.....

one-hit wonder:
Josie Cotton - 'Johnny, Are You Queer?'

The Bloodhound Gang

These also somehow qualify:
Insane Clown Posse
Jimmie's Chicken Shack

Do Spinal Tap count?

Spinal Tap, absolutely, didn't think about them.
And I personally like their music a lot better than some of the stuff on the radio.
And I debated putting GWAR on the list, but for some reason decided against it. Since someone else mentioned them, we'll count them--they can definitely be funny in a *sick, sick* way....
I actually have never listened to ICP or JCS, my brother keeps recommending JCS to me.

Some of Barenaked Ladies' earlier stuff would fit; the proportion of "wacky" stuff has declined over the years.

The Darkness I mean anything released by Buddyhead actually I can't think of any that doesn't resemble baiting. Well, TMBG, Ween, Weird Al, etc. The Scissor Sisters debut makes me laugh in lots of parts, though I don't think of them as "humor rock."

The Bonzo Dog Band
The Rutles

You guys make me feel ancient!

Tenacious D

Captured by Robots

Spinal Tap, Tenacious D, along with Dayglo Abortions, Vandals and Ludichrist/Scatterbrain definitely!

For weird, tongue-in-cheek infused rock you might include...

Clutch
Frank Zappa
Devo
Max Webster
Primus
Soul Coughing

I'd say their progression is somewhat similar to that of their fellow Canadians Moxy Fruvous -- a immensely popular debut that owes its popularity to more 'novelty' type songs, a much more somber followup, a more eclectic third album that falls somewhere in between and divides fans, a live album, culminating in their most recent full-length albums that I can't bring myself to listen to...

Mr. Bungle

What's fun about Spinal Tap and The Rutles is that the songs are more satirical in nature, rather than "LOL" funny. If one heard them out of context on the radio, it would be hard to distinguish them as "humorous".

Definitely would have to add Darkest of the Hillside Thickets to the list. Nothing better than some good Cthulhu humor

Dredging dim memory I think here is the place to mention England's "Macc Lads".

Very crude; definitely not PC; often quite funny, nonetheless.

http://www.macclads.co.uk/images/history/w_hist3.html

Worm Quartet ( www.wormquartet.com has some songs for download)
Mindless Self Indulgence

Check out Wally Pleasent's "I was a Teenage Republican" for a good time.

Forgot to add Cake and Fountains of Wayne as bands not averse to using humor in their songs.

Dead Kennedys are hilarious.

Basically, most everything by SCATTERBRAIN fits this category, not just the one song you name

I find some of Ben Folds' stuff can be funny, if kinda darkly funny.

Not LOL funny, but Weezer can be a little "odd" at times.

The Beat Farmers
Southern Culture on the Skids
The Squirrels
Dread Zeppelin