jmc

Since the denominator of $\frac{3}{16}$ is $2^4$, we multiply numerator and denominator by $5^4$ to obtain $$\frac{3}{16}=\frac{3\cdot 5^4}{2^4\cdot 5^4}=\frac{3\cdot 625}{10^4}=\frac{1875}{10^4}=\mathrm{0.1875.}$$ The digit in the thousandths place is $\boxed{7}$.