  The aim of this work is to publicise some computational results involving
tables which contain $\mathcal{A}$-annihilated monomials, excluding square
classes, in the Dyer-Lashof algebra and $H_*QS^0$; our computations go up to
dimension $1.1\times 10^7$ but the tables in this paper only announce results
up to dimension $2^{17}=131072$ and full tables would be available upon
request. The theoretical background for our computations is provided by work of
Curtis \cite{Curtis} and Wellington \cite{Wellington} on the
$\mathcal{A}$-module structure of the Dyer-Lashof algebra as well as $H_*QS^0$.
It seems to us that there is a workable algorithm to do these computations
which we plan to announce in a future work, partly to avoid making this paper
longer than it is. We hope to receive feedback from the experts on these
computations and make our algorithm available as soon as we can. We hope that
these tables provide a source for researchers in the field, as well as a pool
of data to analyse the behaviour of these sequences, their distributions and
other asymptotic behaviours. The problem of computing spherical classes in
$H_*QS^0$ as well as the symmetric and non-symmetric hit problems have been our
main motivations to pursue this project.
