Can a 7805 ever oscillate?
It's quite challenging to make them misbehave to the point of oscillation in practice, especially with light loads. Try heavy loading, no input capacitance at all and some source inductance, but I offer no guarantees. You can tell as oscillation approaches you'll see reduced phase margin, meaning overshoot/undershoot on load or line changes.  

Based on simulation, it looks like something like 500uH with a 0.5A load will be close to oscillation typically. This is a pretty pathological arrangement. The plot below is with 400uH and a 0.55A load which decreases* to 0.5A at t=100usec



* the decrease is done with a tanh function over a period of about 1usec, so not quite an ideal switch opening.  

This comment does not extend to other types of regulators (especially LDOs) which are easy to make oscillate. And, of course, in real circuits we prefer to live on the side of "guaranteed to be stable" rather than "guaranteed to oscillate", at least for voltage regulators. The opposite would be true of things that are intended to oscillate- as the old saw says "amplifiers oscillate, oscillators don't". 

Edit: I tried a couple quick tests- with a relatively large input capacitor at the regulator (1uF) it shows low level (2.5mV p-p) oscillation at about 8kHz. Much below that it lowers the frequency of oscillation but magnitude remains high. Adding only a capacitor to the output with the input inductor present reduces the stability- 5-10nF is enough to make it oscillate with 400uH inductance on the input and a 0.5A load.