We consider a class of stochastic damped semilinear wave equations, in the small-mass limit. It has previously been established that the solution converges to the solution of a stochastic semilinear heat equation. In this work we exhibit strong and weak rates of convergence in this Smoluchowski--Kramers approximation result. The rates depend on the regularity of the driving Wiener process. For instance, for trace-class noise the strong and weak rates of convergence are $1$, whereas for space-time white noise (in dimension $1$) the strong and weak rates of convergence are $1/2$ and $1$ respectively.