Parallel Anthology Track 54
Track 54: Ernest Phipps and his Holiness Singers - Shine On Me Bluebird 5540-A; Recorded in Bristol, Tennessee, 29.11.1928
No image of record label available. Harry Smith's Anthology Booklet assigns authorship to "E Phipps", although www.78discography.com does not assign authorship.
The "about" section of this YouTube clip of a recording of a different song by Phipps and his singers  quotes from the "Encyclopedia of Appalachia" that Ernest Phipps died in 1963.
This would imply that this composition is in copyright until 1st January 2034.
However, according to the Old Weird America blog:
Some verses of “Shine on me”, sung on the Anthology by Ernest Phipps & His Holiness Singers come from of a 19th century hymn called “Maitland” by George N. Allen which was sung with the words of “Must Jesus Bear The Cross Alone”, a text by Thomas Sheperd. But the chorus (“Shine on me” ) comes from a popular african-american spiritual “Let the light from the lighthouse shine on me” that was recorded many times during the last century.
According to cyberhymnal.org, “Must Jesus Bear The Cross Alone” has word written by Thomas Shepherd (died 1739: ) and music written by George N Allen (died 1877: ). This hymn is therefore public domain.
Let Your Light Shine on Me by Blind Willie Johnson recorded in 1929: Streamable from the Old Weird America blog. The liner notes for a Folkways album containing the same version of this song make no claim to arrangement credit for this version.
Must Jesus Bear This Cross Alone by Sam Cooke recorded in 1956 (according to this site): Spotify According to this site, Sam Cooke has an arrangement credit for this version. According to wikipedia, Cooke died in 1964, so this version is in copyright until 1st January 2015.
The Old Weird America blog provides streams of several other versions whose copyright status is not yet determined, except for the Lead Belly version, which according to the liner notes for a Folkways album containing the same version of the song has an arrangement credit for "Huddie Ledbetter/TRO-Folkways Music, Inc., BMI".
Blog tracks are:
Shine on Me by the Wiseman Sextette
recorded in 1923 according to Old Weird America, which is confirmed by 78discography.com
Record label calls this a "Negro Spiritual". No arrangement credit.
- Shine on Me by Bryant’s Jubilee Quartet recorded in 1928
- Shine on Me by the Fisk Jubilee Singers recorded in the 1930s
- Shine on Me by the Cliff Carlisle Quintet recorded in the 1930s
- Shine on Me by the Swan Silverstones recorded in the 1950s
- Shine on Me by the Kings of Harmony recorded in the 1950s
- Shine on Me by the Soul Stirrers recorded in the 1950s
- Shine on Me by the Reverend Gary Davis recorded in 1954