  In general relativity, all black holes in vacuum are described by the Kerr
metric, which has only two independent parameters: the mass and the spin. The
unique dependence on these two parameters is known as the no-hair theorem. This
theorem may be tested observationally by using electromagnetic or
gravitational-wave observations to map the spacetime around a candidate black
hole and measure potential deviations from the Kerr metric. Several parametric
frameworks have been constructed for tests of the no-hair theorem. Due to the
uniqueness of the Kerr metric, any such parametric framework must violate at
least one of the assumptions of the no-hair theorem. This can lead to
pathologies in the spacetime, such as closed timelike curves or singularities,
which may hamper using the metric in the strong-field regime. In this paper, I
analyze in detail several parametric frameworks and show explicitly the manner
in which they differ from the Kerr metric. I calculate the coordinate locations
of event horizons in these metrics, if any exist, using methods adapted from
the numerical relativity literature. I identify the regions where each
parametric deviation is unphysical as well as the range of coordinates and
parameters for which each spacetime remains a regular extension of the Kerr
metric and is, therefore, suitable for observational tests of the no-hair
theorem.
