Percent-illumination of crescent moon and its naked-eye visibility?
What level of percent-illumination of moon (waxing crescent), given by Stellarium, is enough to make it visible with naked eye, in clear sky?
I would think even 1% would be visible, but am not completely sure. The problem is that the moon is very close to the sun at this point (in the sky, not actually close to the sun). For more information, see: http://en.wikipedia.org/wiki/Lunar_phase
According to Stellarium, even the new moon (at least the one coming up on 10 Jan 2016) has a magnitude of -1.39, which would be visible if it weren't so close to the Sun. The definition of astronomical twilight is that 6th magnitude stars are visible at the zenith when the Sun is 18 degrees below the horizon (ie, 108 degrees away from the stars themselves), but I don't think there's a general formula mapping angular distance from sun to faintest visible magnitude.
The answers I have read are nothing more than unknowing guesses based on what seems logical to the writers. Is there anyone out there that can give an answer based on essential illumination for actual observations. If you say, "That depends on the location of the observers", then Jerusalem would be an excellent location because of the regular watching for the first visible sliver of the new moon from Jerusalem and its surroundings.
To add to Florin's great answer, even the dark part of the moon is visible due to illumination by the earth (has a magnitude of around -3, probably calculated for a new moon/solar eclipse configuration - http://adsabs.harvard.edu/full/1943MNSSA...2...24J, first paragraph of 'brightness of earthshine on moon'). So, the moon is always visible if not in the 'glow' of the Sun.