Electronic – How to measure ESR

This is hard to answer without knowing the what exactly is going on inside the control loop, which is probably not described with much detail in the datasheet. Saying it is "optimized" for 30 mΩ doesn't say what it really needs or what happens when it's not exactly 30 mΩ. Since they make a issue of the output cap ESR, there should be a min/max ESR spec elsewhere in the datasheet.

Hopefully 0 ESR is OK, else it gets inconvenient. Lower ESR is generally better, and certainly closer to ideal, for a capacitor. Capacitor spec sheets therefore often only spec maximum ESR. In that case you either have to get the guaranteed minimum from the capacitor manufacturer, or add deliberate resistance. A better answer is to stay away from chips that specify a minimum output cap ESR to be stable.

That chip is probably old, as ceramic capacitors have come a long way in the last few years. These have much lower ESR than tantalum, and can now approach such capacitances at reasonable cost. Newer chips now make use of the lower available ESR to get better performance instead of requiring the capacitor to have a high resistance. This is a all around better strategy.

This unfortunately depends very much on the application. Most low ESR ('low Z'), low voltage (6.3V and lower) electrolytics (e.g. conductive polymer 'solid' capacitors, power supply output filter capacitors) are characterized at 100kHz. Other bulk capacitors (usually 10-63V) which explicitly give ESR in their datasheets are most often characterized at 10kHz. If your application requires knowledge of ESR, this is what you are looking for.

However, general purpose capacitors seldomly have ESR as a statistic in their datasheets. They often only have the loss tangent (tan \$\delta\$) in the datasheet. This is measured most often at 120Hz, but in some cases also at 1kHz. There is usually no use in trying to measure these capacitors at 10 or 100kHz, because the internal construction prioritizes volumetric capacity over impedance, and the ESL makes the capacitors virtually useless at higher frequencies.

And now I'm only talking about aluminum electrolytics - there are many more nuances to this story when you start looking at other build-ups, like metallized film and ceramic capacitors.

So to answer your question: ALL THE FREQUENCIES.