And here's the overall list of chanties that have turned up, up through the 1870s (again, not accounting for the material in later collections or on recordings by people who sailed in the 1870s). It is based on the sources we've dug up in this thread.

ACROSS THE WESTERN OCEAN (2)
And England's blue for ever"
Black although she be"
BLACKBALL LINE (2)
BLOW BOYS BLOW (2)
BLOW THE MAN DOWN (2)
BONEY (2)
BOTTLE O (1)
BOWLINE (6)
BULLY IN ALLEY (1)
BUNCH OF ROSES (1)
Captain gone ashore!"
Cheerily she goes"
CHEERLY (12)
DRUNKEN SAILOR (1)
FIRE FIRE (2)
GOOD MORNING LADIES (1)
GOODBYE FARE YOU WELL (2)
GOODBYE MY LOVE (1)
GROG TIME (1)
Hah, hah, rolling John" (1)
Hand ober hand, O"
HANDY MY BOYS (1)
HANGING JOHNNY (1)
HAUL AWAY JOE (5)
Heave and she goes, stamp and she goes"
HEAVE AWAY MY JOHNNIES (4)
Heave her away"
Heave him up! O he yo!"
Heave round hearty!"
Heave, to the girls!"
HIGHLAND (2)
Highland day and off she goes"
HILO BOYS (3)
Ho, O, heave O"
HOOKER JOHN (1)
HUNDRED YEARS (2)
Hurrah! hurrah! my hearty bullies!"
Jack Cross-tree,"
John, John Crow is a dandy, O"
JOHNNY BOWKER (2)
Johnny's gone"
Land ho"
LOWLANDS AWAY (2)
Miranda Lee"
MOBILE BAY (1)
MONEY DOWN (1)
MR. STORMALONG (1)
Nancy Bell"
Nancy oh!"
O ee roll & go"
O! hurrah my hearties O!"
Oceanida"
Oh fare you well, my own Mary Anne"
OH RILEY (1)
Oh Sally Brown, Sally Brown, oh!"
ONE MORE DAY (1)
OUTWARD AND HOMEWARD BOUND (2)
PADDY DOYLE (1)
PADDY LAY BACK (1)
PADDY ON THE RAILWAY (4)
Pull away now, my Nancy, O!"
REUBEN RANZO (4)
RIO GRANDE (3)
ROUND THE CORNER (3)
RUN LET THE BULGINE RUN (1)
SACRAMENTO (1)
SAILOR FIREMAN (1)
SALLY BROWN (3)
Sally in the Alley"
SANTIANA (5)
SHENANDOAH (6)
SLAPANDER (1)
STORMALONG JOHN (1)
STORMY (7)
STORMY ALONG (1)
TALLY (2)
Time for us to go!"
TOMMY'S GONE (1)
Walk away"
WALKALONG SALLY (1)
When first we went a-waggoning"
WHISKEY JOHNNY (6)
Whisky for Johnny!"

Of these, the most cited shanties have been:

1. CHEERLY (12 times)
2. STORMY (7)
3. BOWLINE (6) / WHISKEY JOHNNY (6) / SHENANDOAH (6)