In this paper, we investigate ${\mathrm S}$-hypercyclic and ${\mathrm S}$-chaotic strongly continuous $n$-parameter semigroups of operators on finite-dimensional spaces, where $n\in {\mathbb N}$, ${\mathrm S}$ is closed subset of field ${\mathbb K}\in \{{\mathbb R},\ {\mathbb C}\} $ and ${\mathrm S} \setminus \{0\} \neq \emptyset $. We also provide extension of Oxtoby--€“Ulam theorem for multiparameter ${\mathrm S}$-hypercyclic strongly continuous semigroups of operators on arbitrary Fréchet spaces.