  Sharp Besov regularities in time and space variables are investigated for
$\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to
the stochastic heat equation driven by space-time white noise. Existence,
H\"{o}lder continuity, and Besov regularity of local times are established for
$u(t,x)$ viewed either as a process in the space variable or time variable.
Hausdorff dimensions of their corresponding level sets are also obtained.
