Anything by Rudy Vallee

Lazing On a Sunday Afternoon & Seaside Rendezvous - Queen (both on A Night at the Opera)

Thank You Boys - Jane's Addiction

Bow Tie Daddy - Mothers of Invention

The Real Tuesday Weld specializes in "Antique Beat" which seems like it's what you're looking for. He has a bunch of animated youtube videos which are a riot as per:

http://www.youtube.com/watch?v=NgOQuLKeX7g

Seek out the Carter Family, in particular a little ditty by the name of "Single Girl, Married Girl", recorded in 1927 (for a phonograph, indeed). "Single girl, single girl, she goes to the store and buys, oh, she goes to the store and buys. Married girl, married girl, she rocks the cradle and cries, oh, she rocks the cradle and cries." Very affecting.

http://www.youtube.com/watch?v=Ub6Bj08X2PA

Maybelle Carter is one of the great guitar players as well. Check her out on this clip of "Gold Ball and Chain": http://www.youtube.com/watch?v=A8szdi1D7OE

Another 1920s icon is the great Louis Armstrong, my personal favorite human being of all time. Check out "I'm Not Rough" or "Up a Lazy River".

You could also do a lot worse than the great Hoagy Carmichael, like this version of "Old Buttermilk Sky": http://www.youtube.com/watch?v=fOgrjd9NKF0

I Don't Want To Set The World On Fire by The Ink Spots.

Hmm. Check out the Ghost World movie soundtrack. Plenty of old-timey blues songs, with genuine scratching. I especially like Vince Giordano.

Anything by Josephine Baker is fabulous too -- my favourite is "C'est Lui."

And beyond that, not really sure what you're looking for? If you're looking for modern artists who've done songs in early 20th century style, try Squirrel Nut Zippers (although their later stuff, which isn't as good, smacks more of the 50s) or Top Shelf Jazz. Tom Maxwell is a former member of the SNZ and he has some solo work that's fun.

genuine: anything by Django Reinhart.

fake but real-enough-sounding record player 'pops'/scratches etc: RjD2's 'Ghostrider'

The Devil Makes Three are a contemporary band that makes awesome old-timey sounding music (check out "Old Number Seven" and "Shades").

Stephane Grapelli and Django Rheinhardt made wonderful Gypsy jazz in the 30s; very upbeat, swingy and danceable.

Also, check out the Anthology of American Folk Music, an extremely diverse compilation of traditional folk, blues and country made in the late 1920s and early 1930s.

Everything on the O Brother Where Art Thou soundtrack.

"Blowin' Down This Road" by Woody Guthrie
Actually, anything by Woody Gurthrie

Razor Valentine by Thea Gilmore

Roll On Columbia Roll On...
--that is about the Columbia River...here are sample lyrics:

"your power is turning our darkness to dawn...
Roll on Columbia roll on..."

Maybe check out Woody Guthrie tunes

Perhaps "I'm My Own Grandpa" for fun times:
excerpt:

"many many years ago when I was 23 I was married to a widow
who was pretty as could be
this widow had a grownup daughter who had hair of red
my father fell in love with her and soon the two were wed...
I'm my own Grandpa...I'm my own Grandpa...it sounds funny I know
but it really is so, oh, I'm my own grandpa."

old jazz standards are often found in older recordings...Billie Holiday for example

I'm not exactly sure what you are looking for.... good luck!