  The ALE partition functions of a 6d (1,0) SCFT are interesting observables
which are able to detect the global structure of the SCFT. They are defined to
be the equivariant partition functions of the SCFT on a background with the
topology of a two-dimensional torus times an ALE singularity. In this work, we
compute the ALE partition functions of M-string orbifold SCFTs, extending our
previous results for the M-string SCFTs. Via geometric engineering, our results
about ALE partition functions are connected to the theory of higher-rank
Donaldson-Thomas invariants for resolutions of elliptic Calabi-Yau threefold
singularities. We predict that their generating functions satisfy interesting
modular properties. The partition functions receive contributions from BPS
strings probing the ALE singularity, whose worldsheet theories we determine via
a chain of string dualities. For this class of backgrounds the BPS strings'
worldsheet theories become relative field theories that are sensitive to
discrete data generalizing to 6d the familiar choices of flat connections at
infinity for instantons on ALE spaces. A novel feature we observe in the case
of M-string orbifold SCFTs, which does not arise for the M-string SCFT, is the
existence of frozen BPS strings which are pinned at the orbifold singularity
and carry fractional instanton charge with respect to the 6d gauge fields.
