I would like to install fans and and ceiling lights at different locations on a concrete ceiling and not have the wire or the conduit visible. I've seen previous posts recommending a trench as a solution and like this idea. Is there a minimum depth to the trench? (I also tried to find out if the OP of a similar question had successfully pursued any of the solutions offered but that query was deleted for some reason).
If you cut a "trench" in a concrete ceiling, you potentially weaken the strength of the structure. Not a good idea. Better to use already available surface mount wiring channel (trade name "Wiremold" for reference only) and if it bothers you, change your attitude about it. Or hide it beneath some architecturally unnecessary structure, like a faux beam made of stryrofoam and painted to look like the concrete (people sell this for that exact reason). Compromising safety for aesthetics is never a great plan.
The wire would have to be run through conduit, and I don't believe there is a minimum "depth" that the conduit needs to be "buried" in the concrete. So, if using 1/2" conduit, you would need a channel that is at least 1.25" deep by 1" wide. The outside diameter of the conduit is close to 0.75" and you need extra room because I doubt this will be a precision cut, and you need some depth so the concrete patch doesn't immediately crack and fall out. Oh, and you need to fasten the conduit in place so you need to take that into account as well.
Now, that all said, I really doubt you should cut anything into a concrete ceiling. In a multi-unit residence, you likely don't have the legal right to do it, and in any other case, the ceiling is probably concrete because it's structural. And engineer would need to tell you it is safe, in writing.
The normal way to hide electrical in a concrete ceiling would be to add furring strips to the ceiling, run all the electrical, and then build a new ceiling on top of everything. Either that, or hide it with faux wooden beams or similar.
In the ground with the utmost minimal exception: (otherwise it's at least 4", usually 6", if not more)
5: Where solid rock prevents compliance with the cover depths specified in this table, the wiring shall be installed in metal or non-metallic raceway permitted for direct burial. The raceways shall be covered by a minimum of 2 inches of concrete extending down to rock. – adamselectric.coop, PDF
In a concrete ceiling, in a trench after it was poured :