There are a number of proposed ways to compute the stable homotopy groups of spheres. One can rather peculiarly consider stable (co)homotopy of an Eilenberg Maclane spectrum as a generalised (co)homology theory and use the Atiyah–Hirzebruch spectral sequence (in the same way one sometimes uses the Serre spectral sequence knowing information about the $E_{\infty}$ page). Another approach is to use the Adams spectral sequence. Here one takes a so-called Adams resolution of the sphere (it is more sensible to do this with spectra as we then get a genuine free resolution of $\mathbb{Z}/p\mathbb{Z}$ over the Steenrod algebra). One gets a spectral sequence which converges to the p-part of the stable homotopy group. A variant is to do this with some (nice enough I guess) generalised cohomlogy theory which leads to the Adams–Novikov spectral sequence. I have a few different questions:
What are the best results on this? I see here it says that the best known result as of 2007 was up to the 64th stem.
Which method gives the best known results?
In relation to the (classical) Adams spectral sequence one has that the $E_{2}$ terms (mod 2) are given by $\mathrm{Ext}_{A}(\mathbb{Z}/2,\mathbb{Z}/2\mathbb{Z})$. Now this is rather difficult on the face of it to compute as one must find a workable free resolution of $\mathbb{Z}/2\mathbb{Z}$. There is in fact a certain differential graded algebra called that lambda algebra whose cohomology is precisely this. Does anyone know of good source where the details are worked out for this?
Following the last question I wonder if anyone knows any good sources on differentials in the Adams spectral sequence?
[I guess an answer to the last 2 questions is probably just Ravenel's book, but if anyone knows some other fairly readable stuff then that would be more than welcome.]