  We study the pole-skipping phenomenon within holographic axion theories, a
common framework for studying strongly coupled systems with chemical potential
($\mu$) and momentum relaxation ($\beta$). Considering the backreaction
characterized by $\mu$ and $\beta$, we encounter coupled equations of motion
for the metric, gauge, and axion field, which are classified into spin-0,
spin-1, and spin-2 channels. Employing gauge-invariant variables, we
systematically address these equations and explore pole-skipping points within
each sector using the near-horizon method. Our analysis reveals two classes of
pole-skipping points: regular and singular pole-skipping points in which the
latter is identified when standard linear differential equations exhibit
singularity. Notably, pole-skipping points in the lower-half plane are regular,
while those elsewhere are singular. This suggests that the pole-skipping point
in the spin-0 channel, associated with quantum chaos, corresponds to a singular
pole-skipping point. Additionally, we observe that the pole-skipping momentum,
if purely real or imaginary for $\mu=\beta=0$, retains this characteristic for
$\mu \neq0$ and $\beta \neq 0$.
