  We introduce an algorithm to compute the rational torsion subgroup of the
Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a
Magma implementation of our algorithm to a database of curves with low
discriminant due to Sutherland as well as a list of curves with small
coefficients. In the process, we find several torsion structures not previously
described in the literature. The algorithm is a generalisation of an algorithm
for genus 2 due to Stoll, which we extend to abelian varieties satisfying
certain conditions. The idea is to compute p-adic torsion lifts of points over
finite fields using the Kummer variety and to check whether they are rational
using heights. Both have been made explicit for Jacobians of hyperelliptic
curves of genus 3 by Stoll. This article is partially based on the second-named
author's Master thesis.
