  In the wake of interest to find black hole solutions with scalar hair, we
investigate the effects of disformal transformations on static spherically
symmetric space-times with a non-trivial scalar field. In particular, we study
solutions that have a singularity in a given frame, while the action is
regular. We ask if there exists a different choice of field variables such that
the geometry and the fields are regular. We find that in some cases disformal
transformations can remove a singularity from the geometry or introduce a new
horizon. This is possible since the Weyl tensor is not invariant under a
general disformal transformation. There exists a class of metrics which can be
brought to Minkowksi geometry by a disformal transformation, which may be
called disformally flat metrics. We investigate three concrete examples from
massless scalar fields to Horndeski theory for which the singularity can be
removed from the geometry. This might indicate that no physical singularity is
present. We also propose a disformal invariant tensor.
