Forgive my getting serious for a moment....

I did a (very) little research into tunes which actually were used for "Amazing Grace", according to hymnals. As mentioned above and elsewhere, many common meter tunes will work, metrically at least, but these are some which beyond question were used:

• Hephzibah - by John Jenkins Husband. This is the first known pairing of a tune to Newton's text, in A Companion to the Countess of Huntingdon's Hymns (London, 1808). I was unable to find a transcription of the tune, however.
  
• New Britian - composer unknown. The now standard tune, first paired with the text in William Walker's Southern Harmony (1835). The tune appears to be a combination of two tunes, "Gallaher" and "St. Mary", from Columbian Harmony by Charles H. Spilman and Benjamin Shaw (Cincinnati, 1829). "New Britain" itself appeared earlier in Vir­gin­ia Har­mo­ny, by James P. Car­rell & Da­vid S. Clay­ton (Win­ches­ter, Vir­gin­ia: 1831)
  
• Arlington - Thomas Arne, 1762. This tune is used for many hymns.
  
• Belmont - William Gardiner, Sacred Melodies, 1812.        Used for the hymn "By Cool Siloam's Shady Rill."
  
• Glas­gow - Moore's Psalm Sing­er's Pock­et Com­pan­ion, 1756. Used for "Behold! the Mountain of the Lord".
  
• Mar­tyr­dom, aka. Avon or Evening Twilight - Hugh Wil­son, 1800; arranged by Ralph E. Hud­son, cir­ca 1885. Used for multiple hymns.
  
• Pisgah - J.C. Lowry; can be found in Kentucky Harmony and Harmonia Sacra
  Primrose - see Sacred Harp, 47t, where it is used for "Salvation, Oh the Joyful Sound."
  
• Azmon, aka. Denfield - Carl G. Gläser, 1828; ar­ranged by Lowell Ma­son, Mo­dern Psalm­ist, 1839.
  
• Corinth - the tune name is ambiguous, and the likely suspects ("Benediction", "Tantum Ergo" or the "Corinths" in the Sacred Harp [32t] or the Hesperian Harp [43]) don't match the AG text well, having either six lines instead of four or the wrong meter.
  
• Mear - old Eng­lish mel­o­dy, ar­ranged by Aar­on Will­iams (1731-1776). Also in Sacred Harp, 49b.
  
• At the Cross (where I first saw the light) - music by Ralph E. Hudson, 1885, recorded to this tune by Fiddlin' John Carson. (But I think this one is better: At the Cross There's Room, music by Robert Lowry.)
  

Most of these tunes can be found at the Cyberhymnal site: go to the Tunes by Name page and click on the first letter of the tune name.

Other tunes for which I found names only: Lloyd, Evan, Noyes, New Bedford.

I have a recording of AG sung to a different tune by someone like (but not) Doc Watson or Ralph Stanley.