Power supply requirements for a small brushless fan
In particular, I'd like to know what are the risks, if any, of powering it using a very simple AC/DC unregulated power supply (i.e. transformer, diode bridge, filter capacitor).


The cautious approach would be to, since it is unregulated, make sure that the unloaded (open circuit) output voltage of the supply does not exceed the voltage rating of the fan. (Unfortunately, you're not likely to find an absolute maximum voltage rating, as distinct from a nominal supply voltage, for a cheap fan…)

But I wouldn't be surprised if such caution is unnecessary particularly as the motor controller would be built to handle voltage spikes resulting from the motor itself. I can't say from experience.

Of course, if you are planning to run the motor significantly under its rated voltage (to lower the speed) then there's no concern at all here.


  Could the electronics inside the fan, e.g. the controller, misbehave if the power supply is not properly regulated (of course, assuming the unregulated supply voltage output doesn't violate the maximum required by the fan)?


Unlikely. As you've probably already found out, common DC brushless fans are perfectly happy to run on a wide range of voltages and even PWM control; an unregulated supply is nothing compared to that mess. (Though I've heard that a excessively high PWM frequency can stop the controller from working properly.)