A partial geometry $S$ admitting an abelian Singer group $G$ is called of rigid type if all lines of $S$ have a trivial stabilizer in $G$. In this paper, we show that if a partial geometry of rigid type has fewer than $1000000$ points it must be the Van Lint-Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint-Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries.