LDO Regulator Application Note

Ensure the VDD and GND lines are sufficiently robust. If their impedance is too high, noise pickup or unstable operation may result. Connect a capacitor with a suitable value between the VDD and GND pins, and as close as possible to the pins.

In LDO (Low Drop Out) regulators, phase compensation is provided to secure stable operation even when the load current is varied. For this purpose, use a capacitor COUT with good frequency characteristics and proper ESR (Equivalent Series Resistance), value in the hatched area.
If you use a tantalum type capacitor and the ESR value of the capacitor is large, the output might be unstable. Evaluate your circuit including consideration of frequency characteristics.
Depending on the capacitor size, manufacturer, and part number, the bias characteristics and temperature characteristics are different. Evaluate the circuit taking actual characteristics into account.