This is always a topic of much discussion. Also on these forums. For example here in the Nuendo 7 Sticky Post:
My personal opinion - from all I’ve read and researched - is that the frequencies you want to capture are mathematically dependent on the sample rate. Meaning the sample rate has to be twice the value of the highest frequency you want to capture, otherwise those frequencies cannot be recorded. Example: Humans can hear up to 20kHz in their best teenage years, so you need 40kHz sample rate to capture that full spectrum - and play back all frequencies from 0-20kHz without any information loss.
This has been mathematically proven. There are people though who say sample rates above 40kHz increase audio quality. And that also ultrasonic frequencies matter and add subconsciously to the “feeling”. This might be true. But if those ultrasonic frequencies matter, you need the gear to record those super high frequencies above 20kHz. And the gear to play them back, too. And here, if you look at the spec of most mics, even expensive Neumann U87 mics, they record frequencies from 20Hz - 20kHz. They are not spec’d to record past our hearing range. And in my opinion, you don’t need a higher sample rate to capture the output of those mics. Also, many studio monitors and hi-fi speakers cannot reproduce it. Genelec’s 8050B go up to 20kHz. And their big mastering speakers, too. I would assume they know why.
There are microphones, though, that can record up to 100kHz. Those are useful for all kinds of situations. Scientific for example. Recording bats, pitching them down so we humans can hear and study their vocalizations, and those of other animals in the ultrasonic realm. Or sound design, record all kinds of weird frequency content, pitch it way down, and make strange audio experiences audible! This is how Godzilla in the 2014 Godzilla movie was done. Look at the first few minutes: Soundworks Collection
So for me, the sample rate goes together with the signal you record and the gear you have. If you have a mic that records up to 100kHz, you need a sample rate of 192kHz (or 200kHz to be correct) to record it, and later pitch it down for crazy sound design or scientific uses. If you have a crazy mic that records up to 200kHz, then 384kHz sample rate is needed. Otherwise, if you capture “normal” stuff, 48kHz are fine.