The song "Freight Train" by Elizabeth Cotten was my breakthrough fingerpicking song. It's my now favorite uke song and I have memorized it so I can play it any time and anywhere!
The history of this song is pretty interesting. It was written by Elizabeth Cotten when she was a teenager living in rural North Carolina in the early 1900s. Her guitar style was influenced by her banjo-playing brother. She played the guitar in a unique style: she was left-handed but played a right-handed guitar upside down. Cotten gave up playing the guitar when she was a young woman because her pastor asked her to play only sacred music.
Years later, while working in a department store in Washington, DC, she helped find a lost child. It turns out the child, Peggy Seeger, was the daughter of Ruth Crawford Seeger, and the sister of Pete Seeger and Mike Seeger. Mrs. Seeger was so impressed with Cotten, she offered her a job on the spot. While working for the Seegers, Cotten picked up a guitar one day and began playing again. She eventually went on to have a successful career playing the folk music circuit and recording her songs. (Check out the video on YouTube of Cotten playing Freight Train.) Her fingerpicking alternating bass style is known as "Cotten picking," a technique that influenced many guitarists during the folk music boom of the 1960s.
The arrangement I play is by Aaron Keim; the tab is in his Fingerstyle Ukulele Instruction Book, which you can order from his website, The Quiet American. Here's a video that will get you started with the fingerpicking pattern for this song and other similar songs.
Playing fingerstyle ukulele, with information about the songs and where to find the ukulele tablature so that you can play these songs yourself.