I deleted the post Asmyldof says about after a minute since posting (before he made his post) as it wasn't 100% accurate. Just don't wonder what he says about.
It wasn't 100% accurate, but still a valid point that an AC wave on a cap will cost you some that's not measurable past the cap.
but I have a lot of trouble believing they have transients going up to 160V, because if there is TVS protection, the modules should unless isolated by a resistor or something be protecting each other at least to some degree (admittedly, for very very fast transients, probably not as the inductive impedance might create substantial isolation).
I have no trouble at all believing in a car with even 100's of tested devices the main lines still contain spikes past the 100V point.
I also have no trouble believing a small spike like those
can destabilise the input stages of a voltage regulator specced to an absolute maximum of anything below 60V (and even with 60V+ ... protection)
The module test is very nicely quoted by dom0 and as you can see the tests do go up to 150V.
Although the total energy is very low and a simple R-C filter (low R, high C) can usually flatten them off to below 50V, while still allowing sufficient primary current to flow, if you worry about inductances and after-dumps a few more components could be added, such as a TVS and diodes. It will not pass a professional test for production quality modules, but it'll do the job well enough for a one-off.
Also don't be fooled by personal measurements, especially when taken with a digital scope or some similar such. A 160V 0.1us spike can very well not show up at all on your display, or show as a simple 30-ish spike of 0.05 ~ 0.5us. The sampling has its limits and even on analogue scopes the trace of the spike can be so thin that at first glance you miss it entirely. I have spent enough hours in nearly completely dark rooms twisting focus and brightness knobs just to be sure in similar tests and measurements.
As for single stage emitter-follower outputs never oscillating. Yes they sometimes do. It's slightly harder to get them to, but the op-amp stage of a linear regulator will still introduce phase shifting which leads to rush-in-based overcharge and pinch off followed by depletion, turn-on, over-charge, pinch-off, depletion, turn-on, etc. If the C is dampened by a resistance (internal or external) these peaks get smothered to a degree that'll stay compatible with the phase-shift/delay in the error amplification and the oscillation will reduce to (near) nil.
The key in non-oscillation with low ESR capacitances is in the main control circuitry's cleverness with regards to that. (compensation caps on extra pins, feed-forward stabilisations, whatyamaevers)
Sometimes this cleverness is a side product of another intended quality, sometimes it is fully intentional.
With 7805 types, most of them are of rugged enough design (case of luck or research, I do not know, but it's one reason they have been popular for ages) to be very hard to get to oscillate in any significant way. But careful examination of the specific device's (manufacturer dependent) datasheet is never misplaced with regards to ESR requirements.