If $f(x)=\frac{16}{5+3x}$, what is the value of $\left[f^{-1}(2)\right]^{-2}$?
Substituting $f^{-1}(x)$ into our expression for $f$, we get \[f(f^{-1}(x))=\frac{16}{5+3f^{-1}(x)}.\]Since $f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$, we have \[x=\frac{16}{5+3f^{-1}(x)}.\]When $x=2$ this says \[2=\frac{16}{5+3f^{-1}(2)}.\]Solving for $f^{-1}(2)$, we find $f^{-1}(2) = 1$.  Then $[f^{-1}(2)]^{-2} = 1^{-2} = \boxed{1}$.