  Pole-skipping refers to the special phenomenon that the pole and the zero of
a retarded two-point Green's function coincide at certain points in momentum
space. We study the pole-skipping phenomenon in holographic Green's functions
of boundary operators that are dual to massive $p$-form fields and the dRGT
massive gravitational fields in the AdS black hole background. Pole-skipping
points for these systems are computed using the near horizon method. The
relation between the pole-skipping points of massive fields and their massless
counterparts is revealed. In particular, as the field mass $m$ is varied from
zero to non-zero, the pole-skipping phenomenon undergoes an abrupt change with
doubled pole-skipping points found in the massive case. This arises from the
breaking of gauge invariance due to the mass term and the consequent appearance
of more degrees of freedom. We recover the gauge invariance using the
Stueckelberg formalism by introducing auxiliary dynamical fields. The extra
pole-skipping points are identified to be associated with the Stueckelberg
fields. We also observe that, as the mass varies, some pole-skipping points of
the wave number $q$ may move from a non-physical region with complex $q$ to a
physical region with real $q$.
