  This paper provides an in-depth analysis of how computational algebraic
geometry can be used to deal with the problem of counting and classifying
$r\times s$ partial Latin rectangles based on $n$ symbols of a given size,
shape, type or structure. The computation of Hilbert functions and triangular
systems of radical ideals enables us to solve this problem for all $r,s,n\leq
6$. As a by-product, explicit formulas are determined for the number of partial
Latin rectangles of size up to six. We focus then on the study of
non-compressible regular partial Latin squares and their equivalent incidence
structure called seminet, whose distribution into main classes is explicitly
determined for point rank up to eight. We prove in particular the existence of
two new configurations of point rank eight.
