Mention has been made of the "Classic" bases. Lest there should be a misunderstanding, I think that it should be pointed out that it cannot can be said that there is any rigid set of bases called "classic". "Classic" is a somewhat loose term. Some would restrict the term to those bases whch, when folded flat, form a diamond shape: the Kite, Diamond, Fish, Bird and Frog bases. They are what Dr. Solorzano Sagredo called "Deltoids". In the very early days of the modern paperfolding revival there was a tendency by some almost mystical significance. (But Eric Kenneway took a different view and said we should free ourselves from folding from bases. He did!)
A wider concept of "Classic" bases extends the term to Preliminary Fold, Waterbomb Base, Blintz Base, Windmill Base and such exotics as the Blintzed Bird Base and the Stretched Bird Base. Some people might extend the list even further, but does it matter much? A base is only one stage in the folding process and today we are much more liberated from the tyrrany of the bases than we once were.
Turning to another point, it has been asked if there are any books on bases. OUSA's "Origami Basics" has been suggested and I would agree that this is the best collection of bases and basic folding techniques. But it is not the only system. there are many variations in symbols and ways of doing things and none has absolute authority.
I suggest ,however, that there is scope for publication of a wider collection of bases: a collection of bases of of many kinds, both simple and advanced. It would be a kind of resource book in which folders could browse for inspiration. It would be anlogous to a book of standard chess openings. A folder could pick and choose where he wanted or where he found inspiration. Such a book would need to be compiled with taste and with care. There are some monstrous concoctions around which could masquerade under the title of "bases"!
No doubt the collection would include a few models illustrative of what some of the folds could produce, just to relieve the monotony of continuous pages of instructions, but in no way would it be a collection of folding instructions for completed models.
For example: Fred Rohm invented a simple base in the sixties, which he called the "Simplex", a very simple base which he used to surprisingly good and varied effect (Fred would have little to do with the Classic bases!) On a more advanced plane, I can think of the bases that David Brill, Max Hulme and Martin Wall each developed independently in the 197Os and used in their several ways for folding some wonderful animals. So far as I know, they have not been published, except in so far as they may appear as steps in folding instructions for those models which happen to have appeared in print..
Someone would have to write and diagram such a book of bases, and, perhaps even more difficult, they would have to get it published, but I am convinced that if it could be done, it would be a valuable addition to origami literature. Any volunteers?