What are the shanties that mention whaling or have it as a central topic? Perhaps the oldest one of these is one quoted in THE QUID (London, 1832). Doerflinger says that this is the "work of an anonymous author who signs himself only as "a Steerage Passenger," it describes a typical voyage to the Orient in a ship of the Honorable East India Company...." (p. 93) Two (or three?) examples are mentioned. First of all is "Pull away now, my Nancy, O!", with no words given, and then there is this verse:

        "Oh her love is a sailor,
        His name is Jemmy Taylor,
        He's gone in a whaler
        To the Greenland sea!"

and

        "Oh! if I had her,
        Eh then if I had her,
        Oh! how I would love her
        Black although she be."

It's not clear from the context whether these are two different songs or two verses from the same song. Doerflinger gives another old Scottish song on page 307 of his book SONGS OF THE SAILOR AND LUMBERMAN called "Were You Ever In Dumbarton" which has some similar words to "oh! if I had her".

The most obvious example of a shanty that mentions whaling is probably "Reuben Ranzo". W.B. Whall, who went to sea in 1861, gives us a version of this shanty in his book SEA SONGS AND SHANTIES   He says this might have been derived from the name "Lorenzo", "for Yankee whalers took a large number of their men from the Azores, men of Portuguese descent...." (p. 60)

W.L. Alden, in his 1882 article in Harpers, gives a verse and the music for "Randso" (p. 264) which gives the verse about him shipping "on board a whaler".

http://books.google.com/books?id=SsoaAAAAYAAJ&pg=PA284&dq=-+Randso&cd=2#v=onepage&q=-%20Randso&f=false

Doerflinger gives a version by Captain Richard Maitland and one by Captain Patrick Tayluer. Maitland went to sea at the age of 12 in 1869. Tayluer went to sea "about 1885" (p. 323). Maitland's version is the more commonly heard type and Tayluer's gives more "whaling" details but is perhaps later.

Another shanty that mentions whaling is "Tommy's Gone To Hilo". In his book AMERICAN SEA SONGS AND CHANTEYS, Frank Shay says that "Hilo" or Ilo, the southernmost port in Peru, was "something of a message center for whalers." (p. 56). One of the verses found in this shanty mentions that

        "Tommy's gone on a whalin'-ship,
        Oh, Tommy's gone on a damn long trip."   (Hugill, p. 262/'61)

There is an old whaling song known as "The Coast of Peru", which Huntington has from the "Bengal" dated 1832.

        "Our captain has told us
        And we hope it will come true
        That there's plenty of sperm whales
        On the coast of Peru."

He says this song "probably dates back to the last quarter of the eighteenth century when whaling in the Pacific was still new." (p. 4, SONGS THE WHALEMEN SANG)

Two other shanties, which have already been mentioned that refer to whaling are "Shallow Brown"

        "Ship on board a whaler." (Hugill, p. 260/'61)

and "John Cherokee",

        "They shipped him aboard of a whalin' ship." (Hugill, p. 439/'61)

Hugill's "Sister Susan"/"Shinbone Al" has:

        "A whaler's life is no life for me,
        I jumped her an' I left the sea,
        I ran right back to Shinbone Alley." (Hugill, p. 391/'61)

Are there other shanties that mention whaling? I know that some of the whaling songs like "Tis Advertised in Boston" were used as shanties, but I think that's a different category.