From the notes to the Penguin Book (1959):

"To folk-singers, the drowned man in this song is either the Duke of Grantham, or Grafton, or Bedford. Miss Lucy Broadwood suggests (FSJ vol.III, issue 12, 1908, pp.176-9) that he may have been in fact William de la Pole, first Duke of Suffolk who in 1450 was murdered by his political enemies, and his body flung upon the sea-shore at Dover. (Shakespeare writes his version in Henry VI, Part II, Act 4). The 'black was their mourning' stanza occurs also in the ballad of The Death of Queen Jane, presumably being borrowed from Six Dukes if, as seems probable, the latter is the older song. Other 'borrowings' from this ballad appear in a broadside of 1690, called The Noble Funeral of the Renowned Champion the Duke of Grafton who was Slain at the Siege of Cork and Royally Interred in Westminster Abbey. To the tune of, Fond Boy: or, Loves a Sweet Passion. (Printed for Charles Bates at the Sun and Bible in Py-Corner.) [Percy] Grainger recorded three versions in Lincolnshire (FSJ vol.III, 1908, pp.170-4) and Sharp found one (of Yorkshire origin) in the Marylebone Workhouse, London (FSJ vol.V, issue 18, 1914, p.79)." -A.L. Lloyd & Ralph Vaughan Williams.

The Penguin text and tune are in the DT: SIX DUKES WENT A-FISHING. The song was noted by Percy Grainger from George Gouldthorpe of Brigg, Lincolnshire, on July 28th 1906, and first appeared in the Journal of the Folk Song Society, vol.III, issue 12, 1908, page 170.

Steve Roud's Folk Song Index assigns this song Roud number 78.

F.J. Child included an incomplete text, titled The Duke of Bedford, in his Additions and Corrections to The English and Scottish Popular Ballads (vol.V p.298; 1898). This had appeared in Longman's Magazine, XVII, 217, 1890, "sent from Suffolk". Child considered stanzas 5-8 to be "a plagiarism from The Death of Queen Jane, and the remainder to be "so trivial that it is not worth the while at present to assign that piece its own place". He further commented, "I have not attempted to identify this duke of Bedford; any other duke would probably answer as well."

The identity of the victim was further discussed in an article by Mary Rowland, Which Noble Duke? (Folk Music Journal vol.1 no.1, 1965). She considered a number of candidates, including de la Pole (the first Duke of Suffolk); Henry Fitzroy, Duke of Grafton (Charles II's son by Barbara Villiers, and subject of the Noble Funeral broadside of 1690); Lord Tavistock, son of the 4th Duke of Bedford, who died in a hunting accident in 1767; Jasper Tudor, Duke of Bedford and Lord Lieutenant of Ireland (died 1495); John Plantagenet, Duke of Bedford and Regent of France (died in 1435, and buried "betwixt two towers" in Rheims); and George Neville, Duke of Bedford (1457-1483, buried in Yorkshire). She reached no firm conclusion, but rather inclined toward the last possibility.

Bruce Olson has recently published a full text of the Noble Funeral broadside at his Roots of Folk site. I'm not clear from his notes whether this is a collation of the transcription given in Rowland's article and Lucy Broadwood's earlier readings (see below) or taken from another source.

The Duke of Grafton

He also has references to broadside examples under references ZN316 and ZN2703 on his Broadside Ballad Index page.

A version of the song has been discussed in the Forum:

DUKE OF BEFORD - INFO PLEASE  Includes text of The Duke of Bedford noted by Cecil Sharp from William Atkinson (80) at Marylebone Workhouse, London, 9 October 1908, and some comments from Bruce Olson. Atkinson had learned the song from a workmate, Frank Habershon; Habershon was a Sheffield man, and the song had been in his family for several generations, perhaps as far back as c.1780.

The Habershon family version also appears at Lesley Nelson's Folk Music site:

The Duke of Bedford -with midi arrangement.

There is an entry at The Traditional Ballad Index:

Six Dukes Went a-Fishing

A number of versions appeared in The Journal of the Folk Song Society, vol.III, issue 12, 1908, pp.170-179:

• Six Dukes Went A-Fishin': Noted by Percy Grainger from George Gouldthorpe, Brigg, Lincolnshire, July 28th 1906.
• Six Dukes Went A-Fishin': Noted by Percy Grainger from Joseph Leaning, Brigg, Lincolnshire, August 4th 1906.
• Six Dukes Went A-Fishin': Noted by Percy Grainger from Dean Robinson, Brigg, Lincolnshire, July 26th 1906.

These three are accompanied by detailed musical notation made from phonograph recordings.

• The Duke of Grafton: as contributed to the Ballad Society's edition of The Roxburghe Ballads by the then editor, the Rev. J.W. Ebsworth, who had learned it from his father, who had it from his grandfather.
• The Noble Funeral... (referred to above): incomplete text from a damaged broadside in the British Museum, licensed 1690.

The article also contains Lucy Broadwood's speculations on the identity of the Duke, referred to in the "Penguin" notes quoted earlier.

Besides the English examples, there are two sets in Helen Hartness Flanders' Ancient Ballads Traditionally Sung in New England, vol. 3, pp.159-162, which I have not yet seen.

There is a fine parody by the Kipper Family in the DT:

Dover Strand  I assume that this was transcribed by ear by somebody who did not have the benefit of the sleeve-notes or the record label; "Dover Strand" is, of course, a mis-hearing of "Overstrand", the place in Norfolk where the song takes place.  The title is in any case wrong; the song is really called All On the Shore (The Body).  Snuffy has posted detailed corrections to the mistakes in the DT file, plus tune in abc format, in this thread: Kipper Family (+ Lyr corr)