Low dropout regulator circuits, although not all of them, are a
special case where capacitor ESR is more important.

A standard dropout regulator uses a common collector or drain output
transistor and presents a low output impedance. You can still get one
to oscillate with a low ESR output capacitor especially at light loads
where the output impedance is at its highest but it is difficult and
usually trace resistance is enough to prevent it.

A standard low dropout regulator however uses a common emitter or
source output transistor which presents a medium to high output
impedance so the characteristics of the output capacitor become very
important to the frequency compensation and a low ESR just makes
things more difficult. With these you will often find a range of
capacitance and ESR outside of which the regulator will be unstable.

The exception for low dropout regulators are those where a higher
voltage bias supply is used so the output transistor is actually
common collector or drain. These regulators operate like standard
dropout regulators but have low dropout and higher efficiency than low
dropout regulators. Like a standard dropout regulator, they are
tolerant of output capacitor characteristics and ESR.