Is C#'s Random.Next() method thread safe?
There's nothing special done in the Next method to achieve thread safety. However, it's an instance method. If you don't share instances of Random across different threads, you don't have to worry about state corruption within an instance. Do not use a single instance of Random across different threads without holding an exclusive lock of some sort.
Jon Skeet has a couple nice posts on this subject:
StaticRandom
Revisiting randomness
As noted by some commentators, there is another potential problem in using different instances of Random that are thread-exclusive, but are seeded identically, and therefore induce the identical sequences of pseudorandom numbers, because they may be created at the same time or within close temporal proximity of each other. One way to alleviate that issue is to use a master Random instance (which is locked by a single thread) to generate some random seeds and initialize new Random instances for every other thread to use.