  Despite its success in the weak gravity regime, General Relativity (GR) has
yet to be verified in the regime of strong gravity. In this paper, we present
the results of detailed ray tracing simulations aiming at clarifying if the
combined information from X-ray spectroscopy, timing, and polarization
observations of stellar mass and supermassive black holes can be used to test
GR's no-hair theorem. The latter states that stationary astrophysical black
holes are described by the Kerr-family of metrics with the black hole mass and
spin being the only free parameters. We use four "non-Kerr metrics", some
phenomenological in nature and others motivated by alternative theories of
gravity, and study the observational signatures of deviations from the Kerr
metric. Particular attention is given to the case when all the metrics are set
to give the same Innermost Stable Circular Orbit (ISCO) in quasi-Boyer
Lindquist coordinates. We give a detailed discussion of similarities and
differences of the observational signatures predicted for BHs in the Kerr
metric and the non-Kerr metrics. We emphasize that even though some regions of
the parameter space are nearly degenerate even when combining the information
from all observational channels, X-ray observations of very rapidly spinning
black holes can be used to exclude large regions of the parameter space of the
alternative metrics. Although it proves difficult to distinguish between the
Kerr and non-Kerr metrics for some portions of the parameter space, the
observations of very rapidly spinning black holes like Cyg X-1 can be used to
rule out large regions for several black hole metrics.
