The sphere spectrum is the simplest nontrivial spectrum.
The sphere spectrum is the suspension spectrum of the point.
The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
homotopy colimits of simplicial diagrams of $k$-truncated connective spectra for $k \in \mathbb{N}$ are modules over the $k$-truncation $\tau_{\leq k}\mathbb{S}$ of the sphere spectrum.
The sphere spectrum is naturally an E-∞ ring and in fact is the initial object in the (∞,1)-category of ring spectra. It is the higher version of the ring $\mathbb{Z}$ of integers.
Lecture notes include
The Postnikov tower of (localizations of) the sphere spectrum is discussed in
Karol Szumiło, Postnikov tower of the sphere spectrum, Master thesis 2009 (web)
Katja Hutschenreuter, On rigidity of the ring spectra $P_m \mathbb{S}_{(p)}$ and $k o$, (2012) (pdf)
Specifically the 1-truncation of the sphere spectrum (the free abelian 2-group on a single element) is discussed in
The 2-truncation appears for instance in section 3 of