  We study the porosity properties of fractal percolation sets
$E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$,
we obtain dimension bounds for the set of exceptional points where the upper
porosity of $E$ is less than $\tfrac12-\varepsilon$, or the lower porosity is
larger than $\varepsilon$. Our method works also for inhomogeneous fractal
percolation and more general random sets whose offspring distribution gives
rise to a Galton-Watson process.
