Stable homotopy groups of $RP^{\infty}$

The following paper contains a list of stable homotopy of projective spaces in dimensions $\leq 8$:

• A. Liulevicius. A theorem in homological algebra and stable homotopy projective spaces. Transactions of the American Mathematical Society Vol. 109, No. 3 (Dec., 1963), pp. 540-552

JSTOR link

In particular, $\pi^s_5=0$ and $\pi^s_6=\mathbb{Z}/2$.

The middle column of table IV on page 82 of George W. Whitehead's "Recent Advances in homotopy theory" Regional Conference series in mathematics Number 5 lists the groups in dimensions up to 30 (including the 2 quoted by Matthias Wendt).