I had some spare time today, so I wrote a quick program to step an HP6612C power supply over a voltage range and read the resulting output voltage with an Agilent 34410A voltmeter when applied to a fixed voltage regulator. (Uses EZGPIB to control the power supply and voltmeter over the IEEE-488 ports.)
To see if the regulators behave the same way when the voltage starts in regulation and then decreases compared with being out of regulation and increasing, I ran the test with both a start low, ramp up and a start high, ramp low voltage. I've added arrows to the 5V plot to show which path belongs to the increasing and decreasing ramp. The difference between the two paths is the hysteresis. The hysteresis loop is so small in the 10 and 12V regulators that I did not add arrows.
Attached plots are for 5, 10 and 12V standard dropout regulators, in TO-220 or similar metal tab packages. All operate into a 100 ohm fixed resistive load.
A few things of note ...
(1) Note the hysteresis exhibited only by the 5V regulators. Just a trace of hysteresis can be seen on the higher output voltage parts, but it's quite distinctive for both 5V regulators.
(2) Output voltage dropout is quite evident with a sharp knee. (Data is taken in 100 mV steps). Looking at the datapoints, the difference between being in regulation and dropping out of regulation is not more than a 200 or 300 mV.
Leaving the hysteresis area aside, the output voltage for a fixed resistance load looks to be nearly linear with respect to the input voltage. In other words, the regulator operating below the knee is close to a zener type sort of constant voltage drop. The fixed drop in the below threshold regime for TI TL780-12KCS 12V regulator is about 1.5V. This may well be different with a different load impedance, temperature, phase of the moon, etc.
Note that this data is only for one sample part of a limited number of devices and, more importantly, I can't think of a reason - good or bad - why one would intentionally operate a fixed regulator with an input voltage below the regulation knee.
I didn't see signs of oscillation for any of the six regulators, but that's far from a conclusive test. My experience is that the low dropout regulators are much more prone to oscillate under out-of-specification conditions, such as drawing less than the minimum current.