On page 22 of this thesis, it is written that $\pi_*(\Bbb{S})$ is not noetherian. After a bit of thinking and looking online, I haven't found why this is true.
A graded ring is noetherian if its zeroth level is noetherian and if the ideal of elements of positive degree is finitely generated. Since $\pi_*(\Bbb{S})_0=\Bbb{Z}$ is noetherian, it means that $\pi_*(\Bbb{S})_{>0}$ is not finitely generated, which is not very surprising $\pi_*(\Bbb{S})$ being not so well behaved, but why is it true?
If $\pi_*(\mathbb{S})$ were Noetherian, then the ideal of positive-degree elements would be finitely generated, say by homogeneous elements $a_1,\dots, a_n$. It is then easy to see that these $a_1,\dots,a_n$ generate $\pi_*(\mathbb{S})$ as a ring (show this by induction on degree; more generally, for any commutative $\mathbb{N}$-graded ring $A$, $A$ is generated as an $A_0$-algebra by any set of homogeneous generators of $A_{>0}$ as an ideal). But every element of $\pi_*(\mathbb{S})$ of positive degree is nilpotent, so it would follow that $\pi_N(\mathbb{S})=0$ for all sufficiently large $N$. Since this is false, $\pi_*(\mathbb{S})$ cannot be Noetherian.