  The investigation of the dynamical universality classes of quantum systems is
an important, and rather less explored, aspect of non-equilibrium physics. In
this work, considering the out-of-equilibrium dynamics of spinless fermions in
a one-dimensional quasiperiodic model with and without a periodic driving, we
report the existence of the dynamical one-parameter based Family-Vicsek (FV)
scaling of the "quantum surface-roughness" associated with the particle-number
fluctuations. In absence of periodic driving, the model is interestingly shown
to host a subdiffusive critical phase separated by two subdiffusive critical
lines and a triple point from other phases. An analysis of the fate of critical
phase in the presence of (inter-phase) driving indicates that the critical
phase is quite fragile and has a tendency to get absorbed into the delocalized
or localized regime depending on the driving parameters. Furthermore, periodic
driving can conspire to show quantum Kardar-Parisi-Zhang (KPZ)-like
superdiffusive dynamical behavior, which seems to have no classical
counterpart. We further construct an effective Floquet Hamiltonian, which
qualitatively captures this feature occurring in the driven model
