This paper is devoted to studying the behaviors of strongly singular Calderón–Zygmund operators T and their commutators [b, T] generated by T with \(b\in L_{loc}({\mathbb {R}}^n)\) on the Musielak–Orlicz Hardy spaces. The authors obtain the boundedness of T from the Musielak–Orlicz Hardy spaces \(H^\varphi ({\mathbb {R}}^n)\) to the Musielak–Orlicz spaces \(L^\varphi ({\mathbb {R}}^n),\) and from the Musielak–Orlicz Hardy spaces \(H^\varphi ({\mathbb {R}}^n)\) to themselves if \(T^*1=0.\) Meanwhile, the corresponding mapping properties for the commutators [b, T] are also obtained, provided that b belongs to \(\mathcal {BMO}_{\varphi ,u}({\mathbb {R}}^n),\) a non-trivial subspace of \({\textrm{BMO}}({\mathbb {R}}^n).\)
