  We establish a sharp estimate on the negative moments of the smallest
eigenvalue of the Malliavin matrix $\gamma_Z$ of $Z := (u(s, y), u(t, x) - u(s,
y))$, where $u$ is the solution to system of $d$ non-linear stochastic heat
equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents
for the $L^p$-modulus of continuity of the increments of the solution and of
its Malliavin derivatives. These lead to optimal lower bounds on hitting
probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times
\mathbb{R}\}$ in the non-Gaussian case in terms of Newtonian capacity, and
improve a result in Dalang, Khoshnevisan and Nualart [\textit{Stoch PDE: Anal
Comp} \textbf{1} (2013) 94--151].
