I remember reading somewhere that Isaac Asimov once said that there
are only XXX storylines in the world and all fiction is written from
those variations.  How many storylines did he say there were, and what did he
believe them to be?

I didn't find any evidence that it came from Asimov, although he might
have said it at some point.  It is often clained that there are only N
plots in literature/novels, however the number N varies, with 7 being
the most common.

These pages claim that there are only 7 plots:

http://www.bbc.co.uk/arts/books/genre/romance/novelromance/pg2.shtml
http://www.jenniferfallon.com/JFSite/publishing.htm
http://search.barnesandnoble.com/booksearch/isbnInquiry.asp?isbn=1887038043

The following page lists the basic plots, given that there are 1, 3,
7, 20 or 36:

http://www.ipl.org.ar/ref/QUE/FARQ/plotFARQ.html

Search strategy:
only 7 plots novel
://www.google.com/search?q=only+7+plots+novel
only seven plots novel
://www.google.com/search?q=only+seven+plots+novel
Asimov "5 plots"
://www.google.com/search?q=asimov+%225+plots%22
Asimov "five plots"
://www.google.com/search?q=asimov+%22five+plots%22

Please feel free to request clarification.

jeremymiles-ga

Not exactly what I was looking for, but you took me in some
interesting directions that I hadn't forseen.  Thanks!

There is a nice elaboration of the Polti storyline concept here:

http://www.sff.net/people/julia.west/CALLIHOO/ideagen2.htm

This is specifically designed as a creative aid for science fiction
writers. Clicking on each of the "dramatic situations" takes you to a
detailed set of sub-headings related to each "plot." It isn't Asimov,
but it does have an SF slant.