I wish to record some other families of elements in $\mathrm{coker}(J)$ arising from computations of Toda and of Oka of stable homotopy groups at odd primes. These computations aren't merely for small stems (as I incorrectly believed from reading Toda's book and looking at tables in Hatcher or Ravenel texts).
Since the image of the stable J-homomorphism is known explicitly at any prime $p$, we can determine $\mathrm{coker}(J)$ in Toda-Oka range.
As is explained in Appendix B of Milnor-Stasheff, the image of the J-homomorphism in the $k$th stem is a cyclic group $\mathrm{Im}(J_k)$ whose $p$-component $\mathrm{Im}(J_k: p)$ is as follows for an odd prime $p$ (the case $p=2$ is slighly different but is just as easy to describe):
If $\frac{k+1}{2(p-1)}\notin\mathbb N$, then $\mathrm{Im}(J_k: p)$ is zero.
If $\frac{k+1}{2(p-1)}\in\mathbb N$, then $\mathrm{Im}(J_k: p)$ is isomorphic to $\mathbb Z_{p^{r+1}}$ where $p^r$ is the largest power of $p$ that divides $\frac{k+1}{2(p-1)}$.
Toda computed the $p$-component of the $k$th stem for $k<2p^2(p-1)-3$. I won't analyse $\mathrm{coker}(J_k)$ for his range except for one obvious example: If $k=2p(p-1)^2-1$, Toda shows that the $k$th stem has $p$ component $\mathbb Z_p\times\mathbb Z_{p^2}$, which is non-cyclic, and hence $\mathrm{coker}(J_k)$ is nontrivial; in fact $\mathrm{coker}(J_k: p)$ is $\mathbb Z_p$.
Oka in a series of papers, see here and references therein, extended Toda's range and constructed for each $p>3$, some elements (in his notations $\phi$, $\mu$, $\beta$, $\phi_2$) such that $\mathrm{coker}(J_k: p)$ is $\mathbb Z_{p^2}$. In particular, for the $\beta$-elements the degree $k$ is even, and for $\phi$-elements $\frac{k+3}{2(p-1)}\in\mathbb N$, so in these cases the image of $J$ is zero at $p$.
I do not know any examples where $\mathrm{coker}(J_k: p)$ has elements of order $>p^2$, and wonder if this is due to natural limitations of Toda-Oka range, or is there some other explanation of this phenomenon?