  The singly Cabibbo-suppressed decay $D_{s}^{+} \to K^+\pi^{+}\pi^{-}\pi^{0}$
is observed, using a data set corresponding to an integrated luminosity of 6.32
$\rm fb^{-1}$ recorded by the BESIII detector at the centre-of-mass energies
between 4.178 and 4.226 GeV. The first amplitude analysis of $D_{s}^{+} \to
K^+\pi^{+}\pi^{-}\pi^{0}$ reveals the sub-structures in this decay and
determines the fractions and relative phases of different intermediate
processes. The dominant intermediate process is $D_s^+ \to K^{*0}\rho^+$, with
a fit fraction of $(40.5\pm2.8_{\rm{stat.}}\pm1.5_{\rm{syst.}})\%$. With the
detection efficiency based on our amplitude analysis, the absolute branching
fraction for $D_{s}^{+} \to K^+\pi^{+}\pi^{-}\pi^{0}$ is measured to be
$(9.75\pm0.54_{\rm{stat.}}\pm0.17_{\rm{syst.}})\times 10^{-3}$.
