Here are two graphs$^1$ regarding the amount of space debris at different altitudes:
#1: ESA report on debris in 2001
#2: NASA report on debris in 2011
The sharp spikes in the second graph are from the collision between two satellites (Iridium 33 and Cosmos 2251) and a Chinese anti-satellite test. Both occurred between 2001 and 2011 - hence the reason for their non-existence in the first graph.
At what distance above the earth is a satellite most in danger of collision with other objects, including space junk?
Roughly 700-1000 kilometers in altitude. The debris at 800 km was greatly increased after the collision and the Chinese test. There's also a decent spike as we get even higher up, to geostationary orbit. It's crowded there with operational satellites, too.
At what distance does the threat of collision become relatively negligible?
You have to go above geosynchronous orbit (GEO) for it to become negligible, because even at the lowest levels on the graphs, there are still some objects. There isn't any altitude where there's a huge chance of collision, but there's always a chance, unless you get to past High Earth Orbit. Go for the far right side of this diagram:
The red area is near High Earth Orbit.
The main organization that tracks space debris is the NASA Orbital Debris Program Office. This is a US agency; however, other space agencies (such as the ESA) also track space debris. The NASA site is pretty comprehensive, though.
Alternatively, you can just go to the poles. This shows the distribution of objects around Earth:
The objects close to Earth are in Low Earth Orbit; the ring is objects in GEO.
$^1$ As pointed out by Russell Borogove, the graphs differ by quite a lot along the $y$-axis (spatial density). We both suspect that it may be due to different sizes of objects being compared. I'm working on that and trying to figure out if NASA or the ESA has any information. Fortunately, the peaks (even accounting for the Iridium-Cosmos collision and the Chinese anti-satellite test) align at roughly the same altitudes, so the graphs are still applicable.