If we look at typical good listening systems and rooms and habits, it becomes quite evident that 16 bit sampling for the final delivery format is enough.
First, the ambient noise of the room: rooms with 20 dB SPL ambient noise level is exceptionally good. Most rooms we would consider quiet are in the 30-40 dB SPL range. A good stereo system should be able to put out clean uncompressed 105 dB SPL peaks, this is the K85 monitoring level recommendation (85 dB SPL at -20dBFS). Doing simple math shows that with the 0 dBFS hitting 105 dB SPL output level, the theoretical digital noise level of a 16 bit file is at 105-98=7 dB SPL, which is well below the ambient noise of practically ANY room. Using 24 bit sampling would place most of the extra detail so important to audiophiles not only below the room ambient noise, but also below the threshold of hearing. Besides we can in no circumstances discern detail buried under more than 60 dB of louder signal anyway. The very quietest classical recordings that I have heard had bit over 70 dB of DR between peaks and the recoding venue ambient noise, that is over 20 dB less than what 16 bits can accommodate.
I do classical recoding as a hobby, and both converters* I use are "the best in the world" in a broad sense of the word (difficult to establish the truth in this, if there is one, they are damn good anyway). The signal to noise ratio is around 127 dB or 21 bits worth, which seems to be the practical real world limit at the moment. While I wrote that 16 bit sampling is good enough for delivery, I do record at 24/88.2, not 16/44.1. Why?
Using 24 bit sampling gives a lot of extra headroom at the recording stage. Getting the maximum 16 bit quality means trying the set the levels so that peaks hit almost zero, preferably -0.3 dBFS or so, but NEVER CLIP. Nerve wrecking experience especially in live situations where levels can not be exactly set beforehand. Using 24 bits with a world class ADC means almost 30 dB of safety margin for the same end result. In practice I set the levels about 6 dB lower than where I expect the peaks reach.This brings a lot of safety and peace of mind without quality loss. So the 24 bits is more a safety and convenience thing, not something we can actually hear.
Why higher sample rate? This is more for peace of mind, is a client wants a hi-res file afterwards. There is also one slight quality advantage: the best available sample rate converters** are software based (and free, or practically free), so it just might be so that converting a 24/88.2 or 24/96 file into 16/44.1 with those gives a slightly better end result compared to a in-machine 16/44.1 file.
*) Sound Devices 722 field recorder and Prism Orpheus 8-track ADC/CAD converter
**) Izotope SCR bundled with cheap but excellent Audiofile software, or SoX, which is a free command line driven sample rate converter.
Addendum: I made a test file where part of the signal was original 24/96, part 16/44.1 quality edited in at random intervals (converted to 16/44.1 and back to 24/96). I could not hear which parts were 16/44.1. So I am more than skeptical about the "day and night" difference claimed by some.