Or even Bob.
Here’s a guy many of you will not have heard of (unless you are lucky enough to be able to listen to Dr. Demento):
Roy Zimmerman
He does original political satire, with rhymes sometimes rivaling Tom Lehrer. Some of his stuff is pretty funny (maybe less so for conservatives).
Here’s a fan site, with several songs available to play on your computer:
http://www.roysongs.com/sounds/index.htm
Enjoy!
To more directly answer the OP, I think novelty songs are just as available now as they used to be. The distribution mediums have changed, and you may not be as aware of independent novelty songs as you used to be, but they are still there. Instread of Alan Sherman and Tom Lehrer, we have Weird Al Yankovich. For less than super mainstream novelty acts, there are dozens of groups or artists that put out funny material, it’s not not covered by the mainstream music press or played on your normal top 40 stations. One source to find out about these acts is Dr. Demento, but that’s not the only way to find them.
Tony Goldmark is an up-and-coming novelty songwriter who’s had lots of stuff played on Dr. D’s show. Some of it I don’t really like much, but he’s had a couple good ones. He’s not as polished as Weird Al or Tom Lehrer, but some of his stuff really reminds me a lot of Weird Al’s very early recordings.
He’s a 21-year old taking classes at Columbia in Chicago, and he’s taken a couple classes at Second City as well. http://www.tonygoldmark.com/
I personaly like anything by the Arrogant Worms. “Gonna kill the dog next door” is pure Genius! 
Howyadoin,
Hell, whaddya think P. Diddy and Dr. Dre have been putting out for the last 10 years (other than death warrants, that is)…
-Rav
King Missile
The 1920s were the Golden Age of the goofy novelty song:
Horses, Horse, Horses
I’m Just Wild About Animal Crackers
What! No Women?
When I Take My Sugar to Tea
Why Aren’t Yez Eatin’ More Oranges?
My Little Bimbo Down on the Bamboo Isle
T’ain’ Gonna Rain No Mo’
That’s a Lot of Bunk
I Wish I Was in Peoria
Egyptian Ella
My Canary Has Circles Under His Eyes
Yes! We Have No Bananas!
Barney Google
. . . and that’s just off the top of my head . . .
Or check out the '30s and '40s with George Formby.
I’m not sure if they were in the 20’s or not, but another good group was the “Hoosier Hot Shots” who had songs like “I Like Bananas Because They Have No Bones” and “From the Indies to the Andes in his Undies.”
My mother taught me “Yes! We Have No Bananas!” and “Barney Google” back when I was little. But of course you forgot “Does Your Chewing Gum Lose Its Flavor (on the bedpost overnight)?”
Another vote for Wierd Al!
Anybody who likes novelty songs would really appreciate filk! (http://en.wikipedia.org/wiki/Filk) Not all filk songs are (intentionally) funny, but there’s a huge field of Boolean intersection between the two forms. (Any Tom Lehrer song, for instance, is acceptable fare at a filksing.)
Ah, yes, the Worms. Quite good. I’m rather partial to Jesus’ Brother Bob, and Last Saskatchewan Pirate, myself.
I disagree strongly, especially with the “infinitely” part. At the very least, Al is a polished musician, much more so than Sherman or the Capitol Steps. CS are talented, but they’re amateurs, and it does show. Weird Al is at least Sherman’s equal in writing, and Al has acknowledged Sherman as an inspiration.
Since Al doesn’t do political satire AFAIK, his songs aren’t going to have “intriguing implications”. That seems to be at the heart of your objections. Other than that, his lyrics are no more random or dumb than the CS.
That’s simply absurd.
Don’t forget Banana Phone!
I first saw the video for this on his greatest hits DVD. It took me a couple of lines to figure out the song was made up entirely of palindromes.
Very clever indeed.
I think Weird Al’s finest moment was his recent song “Genius In France,” a complex tribute to the great Frank Zappa (complete with Dweezil Zappa on guitar).
Zappa had plenty of material that could be considered “novelty songs,” including “Valley Girl,” “He’s So Gay,” “Don’t Eat the Yellow Snow,” and “Jewish Princess.”
Moxy Fruvous is a terrific Canadian band that combines excellent musicianship and beautiful vocal harmonies with some very witty lyrics. Not all of their stuff is funny, but several songs range from quirky to hilarious. I recommend the album “Live Noise” (a live album, naturally) to get introduced to their unique sound.
When I was a young whippersnapper, it seemed as though every kid in my neighbourhod had K-Tel’s Looney Tunes and/or Goofy Greats.
We won’t see the likes of those days again.
An excellent suggestion. Some of Tom Smith’s stuff, for example, is truly hilarious.
I might also add the world of a cappella, which has its own subsection of novelty songs. Two great online catalogs are Mainely a Capella, which hails — surprise, surprise — from Maine. And on the West Coast, there’s Primarily a Capella.
Some of the best stuff is found on the annual recording of the annual national competition The Harmony Sweepstakes
Highly recommended, and you know the harmonies will be good as well.
I used to have a copy of Tiny Tim’s album, “God Bless Tiny Tim”, and the whole album was nothing but novelty songs.
Then how about “Weird Al” and his contributions to civilisation?