I would like to know, how measurements of the speed of light are conducted these days, especially in the context of the invariance of $c$.
Do all the methods involve mirrors to redirect the photons back to the place where they were emitted from?
If not, could somebody, please, describe the method briefly?
Again, what I am chiefly interested in are attempts to test the invariance, like the Michelson-Morley experiment.
As the highest rated non-selected answer to the question "Has anyone ever measured the one way speed of light perpendicular to the Earth at the Earth's surface?" correctly states, the one way speed of light is an unmeasurable quantity. The only measurement that can be performed is to measure the round-trip speed of light. In other words, you need mirrors.
Since the speed of light is now a defined quantity, scientists no longer technically conduct experiments to measure the speed of light. They instead are conducting experiments to establish the length of a meter. (But it's still a round-trip length that is being measured.)
Suppose you have some independently calibrated concept of a meter and a calibrated timing source, and suppose your observed length differs markedly (not within experimental error) of the calibrated length. You had better check / double check / triple check your calibrations, your lab setup, your wiring, etc. Most likely there's something wrong somewhere. Those superluminal neutrinos weren't superluminal, after all.
The 17th CGPM approved the switch to making the speed of light a defined constant because of 100 years of observations dating back to the Michelson-Morley experiment. Look for a number of similar hakeups in a few months when the next CGPM meets. The metric system is supposedly about to be reborn again.
[...] measurements of the speed of light, especially in the context of the invariance of $c$.
The proposition of wanting to measure the speed of light (in the sense of "signal front speed", a.k.a. "speed of light in vacuum"), comparing "speed of light values" between different trials, is arguably absurd.
Because: quantitative geometric (or kinematic) relations between any particular signal source and (one or more) receiver(s) are not known in any particular trial to begin with.
Consequently, the objective of measurements is the determination of quantitative geometric or kinematic relations between participants in the first place; beginning with the determination of whether and which participants had been at rest to each other in the trial under consideration, or how participants were related differently.
Now, following J. L. Synge ["Relativity. The general theory", p. 108]:
"For us time [duration] is the only basic measure. Length [distance] is strictly a derived concept";
and distance values (i.e. characterizing pairs of participants who were at rest to each other) are expressed as $$\frac{c}{2}~\Delta \tau^{\text{ping}},$$ where "$c$" enters as a conventional, formally invariabe symbol.
Do all the methods involve mirrors [...]
Obviously, the described distance definition is based on considering "pings" (a.k.a. "signal round trips", or "reflections", or "echos").
Since 1983, the meter has been defined to be the length of the path traveled by light in a vacuum in 1/ 299,792,458 of a second. So light in vacuum travels at precisely 299,792,458 m/s by definition. Measurements involving the speed of light are at this point refining precisely how long a meter is, not refining what the speed of light is.
That being said, https://en.wikipedia.org/wiki/Speed_of_light#Measurement lists five different ways of using the speed of light to determine how long a meter is. Any discrepancy between the results of those five methods would be a problem that would need to be investigated, to say the least.
