New results concerning the regularity and exponential decay of the solutions \(u\) of semilinear perturbations of Shubin type equations are obtained. Two cases are studied: i) \(P\) is a partial differential operator with polynomial coefficients, globally elliptic and (formally) selfadjoint and \(u\) is a solution of the equation \(Pu=F(u)+f\) with \(F\) a polynomial function; ii) \(P\) is the harmonic oscillator and \(u\) is a solution of the equation \(Pu=F(x,u,\nabla u)+f\), where \(F\) is a polynomial in \(u\) and \(\nabla\) which coefficients are at most affine functions in \(x\). One proves that if \(f\in S_{\nu}^{\mu} (\mathbb{R}^{n})\), \(\mu \geq 1/2, \nu\geq 1/2\) and \(u\in H^{s}(\mathbb{R}^{n})\) for some \(s> n/2\) in case i) (respectively \(s >1+n/2\) in case ii), then \(u\in S_{\nu}^{\text{max}(1,\mu)}(\mathbb{R}^{n})\). The spaces \(S^{\mu}_{\nu}\) are the \(S\)-type spaces introduced in [I. M. Gelfand, G. E. Shilov, Generalized functions II. Spaces of fundamental and generalized functions. New York and London: Academic Press (1968; Zbl 0159.18301)]. In particular, if \(f=0\), then \(u\) has a superexponential decay of order \(2\) and it can be extended to an analytic function in the strip \(\{z\in \mathbb{C}^{n}; | \operatorname{Im}z | <T\}\) for some \(T>0\). Examples of solutions \(u\) which can not be extended to entire functions are provided. The proof of the theorem is based on the careful study of commutator identities involving multiplication operators and partial differentiation operators combined with perturbative methods in \(S\)-type spaces. The paper contains also an elementary proof for the nonperturbed case \((F\equiv 0)\) and \(\mu=\nu\).