  In this paper, we describe the dynamical symmetries of classical
supersymmetric oscillators in one and two spatial (bosonic) dimensions. Our
main ingredient is a generalized Poisson bracket which is defined as a suitable
classical counterpart to commutators and anticommutators. In one dimension,
i.e., in the presence of one bosonic and one fermionic coordinate, the
Hamiltonian admits a $U(1,1)$ symmetry for which we explicitly compute the
first integrals. It is found that suitable forms of the supercharges emerge in
a natural way as fermionic conserved quantities. Following this, we describe
classical supercharge operators based on the generalized Poisson bracket and
subsequently define supersymmetry transformations. We perform a straightforward
generalization to two spatial dimensions where the Hamiltonian has an overall
$U(2,2)$ symmetry. We comment on plausible supersymmetric generalizations of
the Pais-Uhlenbeck and isotonic oscillators, and also present the possibility
of defining a generalized Nambu bracket within the classical formalism.
