Dixieland music originated in New Orleans in the late 1800's from a mixture of several musical sources including African, French, Spanish and American marching band themes and rhythms. (Coincidentally, the ukulele was also invented in the late 1800's.) Dixieland music and the bands that played it moved north to Chicago and later to New York and the West Coast and has had occasional revivals in the 40's the 50's and the 60's and is probably due for another revival soon. The traditional Dixieland band included a banjo which gives an entree to the ukulele player who can get a banjo uke if he can find the rest of the band. I've found that I prefer the sound of the ukulele to the banjo. For an example listen to the Midnight Serenaders of Portland singing "Crazy 'Bout My Baby"
Even if you don't have a banjo uke or a band, you can still play the vocal accompanied by your ukulele which I think is much better suited to this type of music than is the guitar. If you want the full banjo sound, you can get a plectrum or tenor banjo and tune it like a baritone uke the so-called "Chicago" tuning. The decade of the twenties is called the "jazz age" and the jazz of that era was heavily influenced by Dixieland. The twenties was also the decade of the greatest popularity of the ukulele showing the excellent fit of the ukulele to the music. Check the "music" page and the "uke news" page of this website for songs of the twenties and earlier that can be played with a Dixie beat good examples being "After You've Gone" and "Way Down Yonder In New Orleans". A good introduction to this music can be had by searching youtube.com under "Dixieland Jazz Vocal" and also look at the clips of the Blue Moon Quartet.
The Ukulele Guide Open Mic Songbook shown on our Song Books section has a lot of good Dixieland songs and includes the ukulele chord diagrams and the lyrics under the score. The Dixieland Fake Book C edition Vol 1, available from elderly.com has a great selection of Dixieland songs but most of the songs don't have the words and those that do have the words at the bottom of the page instead of in the score which shows the notes of the melodies and the names of the chords but does not have the chord diagrams.
Dixieland music and the musicians who play it have always been a bit above the mass audience and the musicians seem to play as much for their own enjoyment as for the audience. As a result you will have to do a bit of looking to find the records and sheet music that will enable you to play it. Youtube.com and other internet sites are a good place to listen to some of the masters and you can get the chord names out of a "fake book" some of which have as many as 1200 songs. After you draw out the diagrams for the chords for a while you will have them memorised. Until then you will have to get a good book of ukulele cord diagrams. The song book section of this website has the ukulele chords and sheet music for "After You've Gone" one of the all-time greatest Dixieland songs. The MUSIC section of this website has an mp3 recording of yours truly singing "After You've Gone" with his ukulele. For an interesting example of vocal plus ukulele banjo listen to Sharon Davis and Kevin St. Laurent of the Killer Dillers singing "Nickleodeon" Listen also to the Blue Moon Quartet who have a number of classic Dixieland numbers on youtube.com including "Bourbon Street Parade" (actually a medly of classic New Orleans songs featuring an amplified soprano uke with a pickup playing the part usually given to the banjo and sounding great).
The internet provides only a very small part of the thousands of Dixieland songs and records out there and this page and this website will guide you to some of the best. Ebay.com has listings of 33.3 rpm records for low low prices and garage sales can be a source if you know what to look for. If you go to estate sales, be sure to look in the piano bench for sheet music. Jazzology.com carries the biggest collection of Dixieland and New Orleans music on CD. Even people who are good at reading music will have trouble decoding these songs if they have not heard them so getting and listening to recordings is mandatory.
To get started theere are several ways to search youtube.com and the internet relevant music. Searching for "Dixieland" and "trad jazz" will get you started and of course try Louis Armstrong which will get yoy thousands of songs. Not so well known to the casual listener is Eddie Condon who recorded and led many bands from the 20's through the 60's. A number of the clips have vocals but they usually wait until halfway through the clip to start singing. Eddie started out playing a 4-string plectrum banjo but had to switch to tenor guitar in the thirties when the banjo craze ended. He made many recordings and on some of them you can hear Red McKenzie playing a comb with a small piece of newspaper over it, an instrument which sounds far better than a kazoo. But he also can be seen and heard playing a megaphone-kazoo on "St. Louis Blues" with the Mound City Blues Blowers. (Or he may have tissue paper attached to the megaphone).
The best and largest source of classic Dixieland and popular music is jazzology.com They have many inexpensive sampler CDs with songs by many of the greatest performers so you can find the ones you like and want to order. The sampler CDs are great listening on their own also.
As explained elsewhere on this site, a tenor banjo can be strung with nylon strings and tuned like a baritone ukulele. Here is a video of the Jug Band, wihch features such an instrument, playing "Rag Moma Rag".