  For $(t,x) \in (0,\infty)\times\mathbb{T}^D$, the generalized Kasner
solutions are a family of explicit solutions to various Einstein-matter systems
that start out smooth but then develop a Big Bang singularity as $t \downarrow
0$, i.e., curvature blowup along a spacelike hypersurface. The family is
parameterized by the Kasner exponents $\widetilde{q}_1,\cdots,\widetilde{q}_D
\in \mathbb{R}$, which satisfy two algebraic constraints. There are heuristics
in the mathematical physics literature, going back more than 50 years,
suggesting that the Big Bang formation should be stable under perturbations of
the Kasner initial data, given say at $\lbrace t = 1 \rbrace$, as long as the
exponents are "sub-critical" in the following sense:
$\mathop{\max_{I,J,B=1,\cdots,D}}_{I < J}
\{\widetilde{q}_I+\widetilde{q}_J-\widetilde{q}_B\}<1$. Previous works have
shown the stability of the singularity under stronger assumptions: 1) the
Einstein-scalar field system with $D = 3$ and $\widetilde{q}_1 \approx
\widetilde{q}_2 \approx \widetilde{q}_3 \approx 1/3$ or 2) the Einstein-vacuum
equations for $D \geq 39$ with $\max_{I=1,\cdots,D} |\widetilde{q}_I| < 1/6$.
We prove that the Kasner singularity is dynamically stable for \emph{all}
sub-critical Kasner exponents, thereby justifying the heuristics in the full
regime where stable monotonic-type curvature blowup is expected. We treat the
$1+D$-dimensional Einstein-scalar field system for $D \geq3$ and the $1+D$
dimensional Einstein-vacuum equations for $D \geq 10$. Moreover, for the
Einstein-vacuum equations in $1+3$ dimensions, where instabilities are in
general expected, we prove that all singular Kasner solutions have stable Big
Bangs under polarized $U(1)$-symmetric perturbations of their initial data. Our
results hold for open sets of initial data in Sobolev spaces without symmetry,
apart from our work on polarized $U(1)$-symmetric solutions.
