  The layered perovskite Sr$_2$RuO$_4$ is a most intensively studied
superconductor, but its pairing mechanism, which is often coupled intimately
with magnetic fluctuations in correlated materials, is still an open question.
Here we present a systematic evolution of the Seebeck coefficient in Co- and
Mn-substituted Sr$_2$RuO$_4$ single crystals, in which ferromagnetic and
antiferromagnetic glassy states respectively emerge in proximity to the
superconducting phase of the parent compound. We find that the Seebeck
coefficient $S$ divided by temperature $T$, $S/T$, shows a maximum near
characteristic temperatures seen in the irreversible magnetization $M_{\rm ir}$
in both of the Co- and Mn-substituted crystals, demonstrating both of the
ferromagnetic and antiferromagnetic fluctuations to enhance the Seebeck
coefficient. Interestingly, $S/T$ increases with lowering temperature in the
parent compound, reminiscent of non-Fermi-liquid behavior, indicating an
essential role of coexisting ferromagnetic and antiferromagnetic fluctuations
for the itinerant electrons in Sr$_2$RuO$_4$.
