Ukulele songs can be found on TV (rarely). I first heard Leon Redbone in about 1976 when he appeared on Saturday Night Live. The next day I went to Hastings Books and records and bought ON THE TRACK, his first album. This was many years before I started playing the ukulele. As is the case with all of his albums, the selections date back to the twenties and thirties and before. Some of the easier songs I could play on my guitar by ear since they were of the three-chord variety but most of them totally baffled me. I thought Leon Redbone must be a some kind of genius or music professor to come up with these things. (He may well be both of these--nobody can find any valid biographical information about him.) Back in 1976, we didn't have the internet or eBay or Amazon.com and the local music stores in Amarillo were pretty much sold out of old sheet music, and I never thought of checking garage sales. Even when you have the music, playing the old Tin Pan Alley tunes on the guitar is quite difficult since they are likely to be written in the key of Eb and have many unusual chords that change frequently. For me however these songs are the most satisfying and I can now play a number of them on the ukulele. Leon Redbone has recorded many albums since 1976, and they are available at reasonable prices an CD. If you look on eBay, they are available for much less on lp, and as I have explained in my book, with good equipment, records usually sound better than CDs. (If you talk to people involved in recording, you will find that this is pretty much uncontroverted.) You can listen to samples of the various Leon Redbone albums on Amazon.com and from the Apple I-Tunes music store. Most of his songs are from the days when ukuleles were much more popular than today and are among the best for ukulele.
Finding the sheet music for the old songs popular up until the forties is much harder than getting the recordings. I have read that there may possibly be a Leon Redbone songbook someday but for now it is only a possibility. If you can find original sheet music, it will often have the uklele chord diagrams on it and if not, you can use a tenor ukulele and play the guitar chords. Leon Redbone must be a fan of Ukulele Ike since he makes the kind of vocal trumpet sounds that Ike made, only his are much more pleasant. Some of the songs I recommend from ON THE TRACK are the following: "Ain't Misbehavin", "Lazybones", "Marie", "Lulu's Back in Town", and "Some of These Days". Leon's second album DOUBLE TIME contains the classic "Nobody's Sweetheart", as well as "Shine on Harvest Moon", "Melancholy Baby", "Shiek of Araby", and "If We Never Meet Again This Side of Heaven". His album from 1978, CHAMPAIGN CHARLIE, has`the title song and "Please Don't Talk About Me When I'm Gone", and "Big Bad Bill Is Sweet William Now". The album RED TO BLUE has the classic "Someday Sweetheart", and "Somebody Stole My Gal" and several other good ones. The album FROM BRANCH TO BRANCH, published in 1981, has "When You Wish Upon a Star" (Originally performed in the movie PINOCCHIO by Ukulele Ike), "A Hot Time In The Old Time Tonight", and "My Blue Heaven". You can investigate the songs on his later albums yourself.
Not only does Leon perform these neat songs, he pretty much sticks to the original style of the era in which they were written and does not attempt to recompose them in some individualistic way, a practice that has turned off a lot of people who have not heard the original version. Even though Leon is using a guitar instead of a ukulele, it is an acoustic guitar and when played up on the higher frets isn't too different at that. In 1988, Leon recorded an album of country songs called NO REGRETS, backed up by some of the great Nashville instrumentalists. This is one of his best--not to be missed.