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Perhaps it is worth saying that you can cross check with Ravenel's tables at math.rochester.edu/people/faculty/doug/mybooks/ravenelA3.pdf. For example $\sigma^2$ is actually in the 14-stem. Likewise the other dot in the 2nd band is (I think) the element $x_{28}$ is the 28-stem. Hatcher's "essentially" refers to the fact that there are some elements in coker J in the bottom band, for example $\eta^2 \in \pi_2$ that Craig mentions below
Each dot represents a copy of $Z/p$, whilst connected vertical dots are meant to represent a non-trivial extension (e.g. at $p=2,\eta^3$ corresponds to a $\mathbb{Z}/8$)
There are some nice pictures on Alan Hatcher's webpage: math.cornell.edu/~hatcher/stemfigs/stems.html. In particular the bottom forms the image of the $J$ homomorphism, and you can see how large the coker J is!
I don't know an example in spectra, but if you are willing to work in the $K(n)$-local category then I think the Morava $E$-theory spectrum is an example. Strickland has shown that $D(E_n) = F(E_n,L_{K(n)}S^0) = \Sigma^{-n^2}E_n$, which in turn can be used to show that the natural map $E_n \to D^2E_n$ is an equivalence. In the $K(n)$-local category $X$ dualisable is equivalent to $E^{\vee}_*(X):=\pi_*L_{K(n)}(E \wedge X)$ finitely generated. But I don't think $E^{\vee}_*(E) = \text{Hom}^c(\mathbb{G}_n,E_*)$ is finitely generated
