There are thousands of functions, so a simple list isn't going to work. For specific topics, you can search in documentation (for example search for "decomposition" or "matrix") usually there are tutorials as well that the documentation will point to. Another good source often is mathworld . Hope that helps!

Take, for example, NDSolve. The Documentation Center page for that asserts: "With the default setting Method->Automatic, the Wolfram Language will automatically try to pick the best method for a particular computation."

This strongly suggests that Method -> Automatic does pick one and the same method for all differential equations, but instead that the method may depend on the form of the equation(s), tests to see whether a stiff system is involved, etc., and also that the method is changed dynamically as the integration proceeds.

In fact, if you look at the SomeNotesOnInternalImplementation page cited in a previous answer, you'll read that, for ODEs, NDSolve "by default uses an LSODA approach, switching between a non-stiff Adams method and a stiff Gear backward differentiation formula method."

With Mathematica being a proprietary product, I hardly expect Wolfram Research to reveal much more than that!

Given a choice of options, I can always try all of them and see which one gives the same result as Automatic. It wouldn't be giving anything away to reveal which option choice was used.

"Given a choice of options, I can always try all of them and see which one gives the same result as Automatic. It wouldn't be giving anything away to reveal which option choice was used.":

But the choice will depend upon the particular differential equation, so still the documentation could not tell you in any general way which of the non-stiff Adams method or the stiff Gear backward differentiation formula is used for the explicit, or default, option Method -> Automatic.

Moreover, even if you find through experimentation that an Adams method was used, it won't tell you which order Adams method; for that you'd have to do additional experimentation (including, quite likely, implementing the particular order method).

In principle, Mathematica could be enhanced with an option that, at evaluation time, reports the choice. But that's a different level of functionality from what Mathematica currently has, and it could unduly complicate the underlying code, with unwanted effects on efficiency.