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We establish exact, dimension-dependent, spatio-temporal, uniform and localmoduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (L-KS)SPDEs and for (2) the time-fractional stochastic partial integro-differentialequations (SPIDEs), driven by space-time white noise in one-to-threedimensional space. Both classes were introduced---with Brownian-time-typekernel formulations---by Allouba in a series of articles starting in 2006,where he presented class (2) in its rigorous stochastic integral equationsform. He proved existence, uniqueness, and sharp spatio-temporal H\"olderregularity for the above two classes of equations in $d=1,2,3$. We show thatboth 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 thegradient in both cases. This is unprecedented for SPDEs driven by space-timewhite noise. Our results on exact moduli show that the half-derivative SPIDE isa critical case. It signals the onset of rougher modulus regularity in spacethan both time-fractional SPIDEs with time-derivatives of order $<1/2$ and L-KSSPDEs. This is despite the fact that they all have identical spatial H\"olderregularity, as shown earlier by Allouba. Moreover, we show that the temporallaws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs tothe Houdr\'e-Villa bifractional Brownian motion, yielding a Chung-type law ofthe iterated logarithm for these SPDEs. We use the underlying explicit kernelsand spectral/harmonic analysis to prove our results. On one hand, this workbuilds on the recent works on delicate sample path properties of Gaussianrandom fields. On the other hand, it builds on and complements Allouba'searlier works on (1) and (2). Similar regularity results hold for Allen-Cahnnonlinear versions of (1) and (2) on compacts via change of measure.

L-Kuramoto-Sivashinsky SPDEs vs. time-fractional SPIDEs: exact continuity and gradient moduli, 1/2-derivative criticality, and laws