  The strong tree property and ITP (also called the super tree property) are
generalizations of the tree property that characterize strong compactness and
supercompactness up to inaccessibility. That is, an inaccessible cardinal
$\kappa$ is strongly compact if and only if the strong tree property holds at
$\kappa$, and supercompact if and only if ITP holds at $\kappa$. We present
several results motivated by the problem of obtaining the strong tree property
and ITP at many successive cardinals simultaneously; these results focus on the
successors of singular cardinals. We describe a general class of forcings that
will obtain the strong tree property and ITP at the successor of a singular
cardinal of any cofinality. Generalizing a result of Neeman about the tree
property, we show that it is consistent for ITP to hold at $\aleph_n$ for all
$2 \leq n < \omega$ simultaneously with the strong tree property at
$\aleph_{\omega+1}$; we also show that it is consistent for ITP to hold at
$\aleph_n$ for all $3 < n < \omega$ and at $\aleph_{\omega+1}$ simultaneously.
Finally, turning our attention to singular cardinals of uncountable cofinality,
we show that it is consistent for the strong and super tree properties to hold
at successors of singulars of multiple cofinalities simultaneously.
