  We study the singularities of varieties obtained as infinitesimal quotients
by $1$-foliations in positive characteristic. (1) We show that quotients by
(log) canonical $1$-foliations preserve the (log) singularities of the MMP. (2)
We prove that quotients by multiplicative derivations preserve many properties,
amongst which most $F$-singularities. (3) We formulate a notion of families of
$1$-foliations, and investigate the corresponding families of quotients.
