symmetric monoidal (∞,1)-category of spectra
The definition of a Gorenstein ring spectrum is motivated by the Gorenstein condition for rings. A Gorenstein ring, $R$, is a commutative Noetherian local ring such that the Ext-group $Ext^{\ast}_R(k, R)$ is one dimensional as a $k$-vector space, where $k$ is the residue field of $R$. This last condition may be restated as the property that the homology of the (right derived) Hom complex $Hom_R(k, R)$ is equivalent to a suspension of $k$.
Thus, a ring spectrum $\mathbf{R} \to \mathbf{k}$ is said to be Gorenstein if there is an equivalence of $\mathbf{R}$-module spectra $Hom_{\mathbf{R}}(\mathbf{k}, \mathbf{R}) \simeq \Sigma^a\mathbf{k}$ for some integer $a$.
A ring, $R \to k$, is Gorenstein if and only if the ring spectrum $H R \to H k$ is Gorenstein.
Examples from representation theory, from chromatic stable homotopy theory and from rational homotopy theory are given in (Greenlees16, Sec. 23).
William Dwyer, John Greenlees, Srikanth Iyengar, Duality in algebra and topology, Advances in Maths
200 (2006) 357-402, (arXiv:math/0510247)
John Greenlees, Homotopy Invariant Commutative Algebra over fields, (arXiv:1601.02473)