According to this answer the LM7805 has no minimum load. The data sheet also indicates that the output voltage should remain between 4.8 and 5.2 V for output currents of between 5 mA and 1 A.
I have a LM7805CA from an unknown manufacturer (I don't recognize the logo, which is an upward-pointing triangle to the left of a downward facing triangle enclosed by lines on all but the bottom side), but the additional markings appear to be "GOOKJ V6" and "MAR 918." (Unfortunately the printing on it is such that it's difficult to read and near-impossible to photograph.)
I've got the output hooked up to a constant-current variable load and the input is around 9 V (I see similar behaviour throughout a 7 V to 12 V input range) from a bench power supply, like so:
The LM7805 is underneath the red alligator clip (you can just see the edge peeking out the top) and test clips you see there lead to my multimeter; I'm taking all voltage readings from that since it's more accurate than the load's display. (The actual output voltage in the piacture is 4.96 V.)
As you can see in the image, with about 170 mA of load the output voltage is within spec. As I increase the load to 200 mA the output voltage increases to around 5.02 V and stays there even up past 500 mA.
However, when I reduce the load to around 120 mA the voltage drops to 4.84 V (still within spec, but barely) and at 80 mA the output voltage drops to 4.72 V. The load won't reliably sink much less than this (I have seen 4.66 V at around 60 mA), but earlier, playing with just an LED and 330 Ω resistor in series between the output and ground (on a breadboard), I saw the output voltage drop even further, to around 4 V.
So what's going on here? Is this device really going out of spec, or is it something to do with the way I'm measuring things?