Explicit formulae are obtained for the distribution of various random partitions of a positive integer $n$, both ordered and unordered, derived from the zero set $M$ of a Brownian motion by the following scheme: pick $n$ points uniformly at random from $[0,1]$, and classify them by whether they fall in the same or different component intervals of the complement of $M$. Corresponding results are obtained for $M$ the range of a stable subordinator and for bridges defined by conditioning on $1 \in M$. These formulae are related to discrete renewal theory by a general method of discretizing a subordinator using the points of an independent homogeneous Poisson process.