  This paper is concerned with two dual aspects of the regularity question of
the Navier-Stokes equations. First, we prove a local in time localized
smoothing effect for local energy solutions. More precisely, if the initial
data restricted to the unit ball belongs to the scale-critical space $L^3$,
then the solution is locally smooth in space for some short time, which is
quantified. This builds upon the work of Jia and \v{S}ver\'{a}k, who considered
the subcritical case. Second, we apply these localized smoothing estimates to
prove a concentration phenomenon near a possible Type I blow-up. Namely, we
show if $(0, T^*)$ is a singular point then
$$\|u(\cdot,t)\|_{L^{3}(B_{R}(0))}\geq \gamma_{univ},\qquad
R=O(\sqrt{T^*-t}).$$ This result is inspired by and improves concentration
results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We
also extend our results to other critical spaces, namely $L^{3,\infty}$ and the
Besov space $\dot B^{-1+\frac3p}_{p,\infty}$, $p\in(3,\infty)$.
