  We report on new measurements of Cabibbo-suppressed semileptonic $D_s^+$
decays using $3.19~\mathrm{fb}^{-1}$ of $e^+e^-$ annihilation data sample
collected at a center-of-mass energy of 4.178~GeV with the BESIII detector at
the BEPCII collider. Our results include branching fractions $\mathcal
B({D^+_s\rightarrow K^0 e^+\nu_{e}})=(3.25\pm0.38({\rm stat.})\pm0.16({\rm
syst.}))\times10^{-3}$ and $\mathcal B({D^+_s\rightarrow K^{*0}
e^+\nu_{e}})=(2.37\pm0.26({\rm stat.})\pm0.20({\rm syst.}))\times10^{-3}$ which
are much improved relative to previous measurements, and the first measurements
of the hadronic form-factor parameters for these decays. For $D^+_s\rightarrow
K^0 e^+\nu_{e}$, we obtain $f_+(0)=0.720\pm0.084({\rm stat.})\pm0.013({\rm
syst.})$, and for $D^+_s\rightarrow K^{*0} e^+\nu_{e}$, we find form-factor
ratios $r_V=V(0)/A_1(0)=1.67\pm0.34({\rm stat.})\pm0.16({\rm syst.})$ and
$r_2=A_2(0)/A_1(0)=0.77\pm0.28({\rm stat.})\pm0.07({\rm syst.})$.
