  Seven different refinements of trapped surfaces are proposed, each intended
as potential stability conditions. This article concerns spherical symmetry,
but each condition can be generalized. Involute trapped spheres satisfy a
similar condition to minimal trapped spheres, which are are strictly minimal
with respect to the Kodama vector. There is also a weaker version of involute
trapped. Outer trapped spheres have positive surface gravity. Increasingly
(future, respectively past) trapped spheres generate spheres which are more
trapped in a (future, respectively past) causal direction, with three types: in
any such causal direction, along the dual Kodama vector, and in some such
causal direction. Assuming the null energy condition, the seven conditions form
a strict hierarchy, in the above order. In static space-times, they reduce to
three inequivalent definitions, namely minimal, outer and increasingly trapped
spheres. For a widely considered class of so-called nice (or non-dirty) black
holes, minimal trapped and outer trapped become equivalent.
Reissner-Nordstr\"om black holes provide examples of this, and that
increasingly trapped differs. Examples where all three refinements differ are
provided by a simple family of dirty black holes parameterized by mass and
singularity area.
