  Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking values
in ${\mathbb{R}}^d$, specified by the following stochastic partial differential
equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial
t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W}(x,t)$ is
an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe (2002)
have proved necessary and sufficient conditions for the ${\mathbb{R}}^d$-valued
process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ to hit points and to have double
points. In this paper, we continue their research by determining the Hausdorff
and packing dimensions of the level sets and the sets of double times of the
random string process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also consider
the Hausdorff and packing dimensions of the range and graph of the string.
