  Many concepts from extremal set theory have analogues for families of
permutations. This paper is concerned with the notion of shattering for
permutations. A family $\mathcal{P}$ of permutations of an $n$-element set $X$
shatters a $k$-set from $X$ if it appears in each of the $k!$ possible orders
in some permutation in $\mathcal{P}$. The smallest family $\mathcal{P}$ which
shatters every $k$-subset of $X$ is known to have size $\Theta(\log n)$.
  Our aim is to introduce and study two natural partial versions of this
shattering problem.
  Our first main result concerns the case where our family must contain only
$t$ out of $k!$ of the possible orders. When $k=3$ we show that there are three
distinct regimes depending on $t$: constant, $\Theta(\log\log n)$, $\Theta(\log
n)$. We also show that for larger $k$ these same regimes exist although they
may not cover all values of $t$.
  Our second direction concerns the problem of determining the largest number
of $k$-sets that can be totally shattered by a family with given size. We show
that for any $n$, a family of $6$ permutations is enough to shatter a
proportion between $\frac{17}{42}$ and $\frac{11}{14}$ of all triples.
