My list includes:
Homer & Jethro
Weird Al Yankovic
Tom Lehrer
Allan Sherman
Smothers Brothers*
Spike Jones (natch!)
*mostly, the Smothers Brothers’ fame was their dialog; IIRC, they published a non-comedy album (The Smothers Brothers Play It Straight).
A few anomalies:
–Dean Martin, on his 60s-70s variety show; halfway through the show he’d sit down while Ken Lane played the piano and sing lines line these:
“Climb upon my knee, Sonny Tufts!!!”
“June is busting out all over;/June had better get a bigger dress!” 
–The Three Stooges
–Laurel & Hardy (remember Way Out West?)
Post your own favorite novelty musicians, singers, songs, and anomalies here. 
Victor Borge
Monty Python
Cool list. Dean and Ken also had “I’m Gonna Sit Right Down and Write My Irish Setter”
Not quite a novelty singer, but definitely an anomaly:
Wesley Willis
I’m a fan of Judge Dread and the Macc Lads. Of course I’m equating novelty with bawdy here… I suppose that there are those who take these guys more seriously than is, well, healthy really.
Allan Sherman is one of my favorite recording artists of all time. I can’t explain it, but wow was that man funny.
Hail to thee, o fat person! You kept us out of war!
Arrogant Worms
The Firesign Theater, especially the "How Can You Be In Two Places At Once… album
Gotta go with Suo Na and say the Arrogant Worms. Also, I’m surprised nobody has said Corky and the Juice Pigs, yet.
The Bonzo Dog Band. The greatest by far.
Spike Jones
Tom Lehrer
Anomalies?
Cream – “Mother’s Lament” is one of the nuttiest songs around.
Procol Harum – “Mabel” “Don’t eat green meat/it ain’t good for you.”
Cole Porter – “Brush Up Your Shakespeare”
TENACIOUS D
They rock my freakin’ socks off!
Tim Cavanaugh, author of such great songs as:
“I Wanna Kiss Her (But She Won’t Let Me)”
“The Piston Song”
“Let’s Drink Up, Chuck and Di”
“99 Dead Babboons”
And Tim Wilson, who has written, among others:
“Garth Brooks Has Ruined My Life”
“First Baptist Bar and Grill”
“Acid Country”
“Help Me Find Jimmy Hoffa”
“This one’s Jake, that one’s Dinah,
There’s big Ned in my recliner.”
Romanovsky & Phillips. “Guilt Trip,” “I’m In Love With My Therapist,” “Sodomy Song” (political AND bawdy).
Spinal Tap
Really?..
Back when I was a college DJ [radio voice]WUNH - Radio Free Durham[/radio voice], we had a comedy show (the Cheese Flavored Show) that I would frequently listen to and occasionally host. A lot of Firesign Theater used to get played. I never got it. There were occasional almost funny moments, but most of it seemed to fall under the category of “I bet this would be funny if I were stoned.”
Twelve replies so far and no one’s mentioned Shel Silverstein yet? I guess the emphasis in the OP was on the musicians/singers and not songwriters, but still – “Boy Named Sue”, “Cover of the Rolling Stone”, “Don’t Give a Dose to the One You Love Most”, “I Got Stoned and Missed It”, “The Unicorn”, et al.?
My own favorites:
[ul]
[li]Alberto y Los Trios Paranoias: the entire brilliant Snuff Rock EP[/li][li]Race Marbles: “Like A Dribbling Fram”[/li][li]The Rutles: The Rutles[/li][li]The Bonzo Dog (Doo Dah) Band[/li][li]Spinal Tap[/li][li]The early National Lampoon Radio Hour troupe’s music parodies, especially “Deteriorata”, “You Put Me Through Hell”, most of Lemmings, “Well Intentioned Blues”, “Methadone Maintenance Man”, “Magical Misery Tour”[/li][/ul]
And a few anomalies? Well, Nick Lowe’s almost always amusing, but “Rollers’ Show”, “I Love My Label” and “Marie Provost” stand out. Likewise, Kirsty MacColl’s wit was one of her many strong points, but “There’s a Guy Works Down at the Chip Shop Swears He’s Elvis” takes it one step further. Translator’s obscure b-side “Ronnie Raygun Blues” is unlike anything else they ever did, and includes the memorable line “I like the way his neck skin wiggles when he chews”.
I second the Messrs. Leher, Sherman and Jones. And I raise you:
• Helen Kane, the “Boop-a-Doop” girl of the 1920s
• Frank Crumit
• Ukulele Ike (Edwards, that is)
• Tiny Tim (crazy as a loon, but always liked him)
Most of my favorites have been mentioned so i’ll toss in my favorite anomalies:
First and foremost, JOHNNY CASH. Shel Silverstein’s “A Boy Named Sue” was mentioned, but that song would’ve gone nowhere if Johnny Cash hadn’t lent his voice to it. And then there’s Cash’s other “novelty” (although I hate using that word to describe music) “One Piece At A Time”.
Chuck Berry’s “My Ding-A-Ling”
And then there’s the Barenaked Ladies (correct me if I’m wrong about the band, but I think they did this one) “If I Had A Million Dollars” which amuses the hell out of me.
And there was a band called Big Daddy that did '50s and early ‘60s style covers of modern songs…brilliant stuff. They did the Beatles’ “Day in the Life” as Buddy Holly and the Crickets would’ve done it, and turned “Girls Just Wanna Have Fun” into a remake of “Duke of Earl”.
For his instrumental piano covers of punk classics such as the Dead Kennedys’ “Holiday in Cambodia” amongst other things.