  Highlighting the use of critical classes, we consider constituents in
Kronecker products, in particular of spin characters of the double covers of
the symmetric and alternating groups. We apply results from the spin case to
find constituents in Kronecker products of characters of the symmetric groups.
Via this tool, we make progress on the Saxl conjecture; this claims that for a
triangular number $n$, the square of the irreducible character of the symmetric
group $S_n$ labelled by the staircase contains all irreducible characters of
$S_n$ as constituents. We find a large number of constituents in this square
which were not detected by other methods. Moreover, the investigation of
Kronecker products of spin characters inspires a spin variant of Saxl's
conjecture.
