I recently started to read about May spectral sequence, which converge to the $E_2$ term of the classical ASS.
At the prime $2$, this is a spectral sequence with $E_1$ page a polynomial algebra on generators $h_{i,j}$ for $i\ge 1, j\ge 0$ with differentials $d_1(h_{i,j})=\sum_k h_{i-k,j+k}h_{k,j}$.
I saw many papers, such as Ravanel's book http://web.math.rochester.edu/people/faculty/doug/mu.html and others, a computation giving quickly the $E_2$-term of the ASS at the range $t-s \le 13$ or so is presented. I also saw many claims like "this is the best way to compute the $E_2$ term by hands", which seems pretty clear from small experiment by hand.
What I want to know though, is how far we are from chasing it completely up to page $k$ for some $k$-s? e.g. is the entire $E_2$ page of MSS known? and if not, how many complete "columns" of the $E_2$-page, consisting of monomials up to given degree in the $h_{i,j}$, are known completely?
