Waxing crescent moon - the young moon
A young Moon occurs between a new Moon and waxing crescent, when the first slivers of illumination appear. There are no real formulas for determining when this will happen. It depends on several factors. The first is the angle of the ecliptic with respect to the horizon. From our perspective on Earth, this means the path the Moon will make across the sky. The higher and more visible it is, the easier it is to see the Young Moon. In addition the first occurrence of a young Moon depends on how clear the sky is. Dust and pollution may interfere with being able to see those first slivers of light. Finally, eyesight plays a role. The clearer your vision, the more illumination you will be able to make out.
If everything aligns in your favor, then you may be able to see a young Moon as little as 24 hours after it was a new Moon. Every additional hour after this will dramatically increase your chance of seeing it. As a final note, the young Moon becomes visible to the unaided eye much earlier at times when the ecliptic is perpendicular to the horizon. This means a time when the Moon pops straight up into the sky, as opposed to hugging the horizon. The record for the earliest claimed sighting of the young crescent Moon was around 19 hours, although most experts are suspicious of any claims of times less than about 24 hours.
If everything aligns against you, then it may take up to 36 hours to first see the young Moon. This occurs when the ecliptic is at a low angle to the horizon, and the Moon moves almost parallel to the horizon as it rises.