TITLE: What are the best known results for the stable homotopy groups of spheres?
QUESTION [17 upvotes]: 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 to deduce it about the $E_{2}$ page). Another approach is to use the Adams spectral sequence. Here one plays off the rigidity of the cohomology of generalised Eilenberg Maclane spectra against this failing in general. This leads to a spectral sequence which converges to the p-part (where $p$ refers to taking cohomology in $\mathbb{Z}/p\mathbb{Z}$) of the stable homotopy group of spheres. A variant is to do this with some (nice enough I guess) generalised cohomology theory which leads to the Adams–Novikov spectral sequence. I think there are probably quite a few other methods which are used to calculate these. My questions are:

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?
What stops us here? Do we simply not yet know the differentials around 64 in the Adams spectral sequence?

REPLY [8 votes]: You could look at this talk: http://neil-strickland.staff.shef.ac.uk/talks/oslo_talk.pdf
About the first third of the talk is a map of various different approaches to the stable and unstable homotopy groups of spheres.  However, several approaches are missing:

Kochman's approach via the AHSS for $\pi_*(BP)$, as in Christian Nassau's answer.
The similar approach that you mentioned, via the AHSS for $\pi_*(H/p)$
The superstable EHPSS, which is essentially the AHSS for the pro-spectrum $\mathbb{R}P_{-\infty}^\infty$ together with the theorem that that pro-spectrum is $2$-adically equivalent to  $S^{-1}$.  (There is also an analog at odd primes using $B\Sigma_p$ in place of $\mathbb{R}P^\infty$.)  This should enable one to build a bridge between Kochman-style calculations and the EHP sequence and the Goodwillie tower, but I'm not aware that anyone has worked that out.

(I should mention that Mark Behrens has independently worked out most of the ideas mentioned in the remainder of the talk that I linked to, with much more success than I had.)