  For integers $n>2$ and $k>0$, an $(n\times n)/k$ semi-Latin square is an
$n\times n$ array of $k$-subsets (called blocks) of an $nk$-set (of
treatments), such that each treatment occurs once in each row and once in each
column of the array. A semi-Latin square is uniform if every pair of blocks,
not in the same row or column, intersect in the same positive number of
treatments. We show that when a uniform $(n\times n)/k$ semi-Latin square
exists, the Schur optimal $(n\times n)/k$ semi-Latin squares are precisely the
uniform ones. We then compare uniform semi-Latin squares using the criterion of
pairwise-variance (PV) aberration, introduced by J.P. Morgan for affine
resolvable designs, and determine the uniform $(n\times n)/k$ semi-Latin
squares with minimum PV aberration when there exist $n-1$ mutually orthogonal
Latin squares (MOLS) of order $n$. These do not exist when $n=6$, and the
smallest uniform semi-Latin squares in this case have size $(6\times 6)/10$. We
present a complete classification of the uniform $(6\times 6)/10$ semi-Latin
squares, and display the one with least PV aberration. We give a construction
producing a uniform $((n+1)\times (n+1))/((n-2)n)$ semi-Latin square when there
exist $n-1$ MOLS of order $n$, and determine the PV aberration of such a
uniform semi-Latin square. Finally, we describe how certain affine resolvable
designs and balanced incomplete-block designs (BIBDs) can be constructed from
uniform semi-Latin squares. From the uniform $(6\times 6)/10$ semi-Latin
squares we classified, we obtain (up to block design isomorphism) exactly 16875
affine resolvable designs for 72 treatments in 36 blocks of size 12 and 8615
BIBDs for 36 treatments in 84 blocks of size 6. In particular, this shows that
there are at least 16875 pairwise non-isomorphic orthogonal arrays
$\mathrm{OA}(72,6,6,2)$.
