  We establish exact, dimension-dependent, spatio-temporal, uniform and local
moduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (L-KS)
SPDEs and for (2) the time-fractional stochastic partial integro-differential
equations (SPIDEs), driven by space-time white noise in one-to-three
dimensional space. Both classes were introduced---with Brownian-time-type
kernel formulations---by Allouba in a series of articles starting in 2006,
where he presented class (2) in its rigorous stochastic integral equations
form. He proved existence, uniqueness, and sharp spatio-temporal H\"older
regularity for the above two classes of equations in $d=1,2,3$. We show that
both classes are $(1/2)^-$ H\"older continuously differentiable in space when
$d=1$, and we give the exact uniform and local moduli of continuity for the
gradient in both cases. This is unprecedented for SPDEs driven by space-time
white noise. Our results on exact moduli show that the half-derivative SPIDE is
a critical case. It signals the onset of rougher modulus regularity in space
than both time-fractional SPIDEs with time-derivatives of order $<1/2$ and L-KS
SPDEs. This is despite the fact that they all have identical spatial H\"older
regularity, as shown earlier by Allouba. Moreover, we show that the temporal
laws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs to
the Houdr\'e-Villa bifractional Brownian motion, yielding a Chung-type law of
the iterated logarithm for these SPDEs. We use the underlying explicit kernels
and spectral/harmonic analysis to prove our results. On one hand, this work
builds on the recent works on delicate sample path properties of Gaussian
random fields. On the other hand, it builds on and complements Allouba's
earlier works on (1) and (2). Similar regularity results hold for Allen-Cahn
nonlinear versions of (1) and (2) on compacts via change of measure.
