  The study of substructures in random objects has a long history, beginning
with Erd\H{o}s and R\'enyi's work on subgraphs of random graphs. We study the
existence of certain substructures in random subsets of vector spaces over
finite fields. First we provide a general framework which can be applied to
establish coarse threshold results and prove a limiting Poisson distribution at
the threshold scale. To illustrate our framework we apply our results to
$k$-term arithmetic progressions, sums, right triangles, parallelograms and
affine planes. We also find coarse thresholds for the property that a random
subset of a finite vector space is sum-free, or is a Sidon set.
