What ML methods exist to categorize signal from noise? Red noise? Spatially correlated noise? Let's say we are given measurements of some sort. 

In many cases, it is safe to assume that noise is white noise, serially uncorrelated, and zero mean with some finite variance. 

But in other cases, "red noise" exists such that the noise is correlated in time. What does one do in this case? Perhaps a Gaussian Process could be used?

What methods exist to characterize spatially-correlated noise?
The basic approach is to model the noise as an appropriate stochastic process (aka random process).  There's an entire subfield of statistics that has studied different kinds of stochastic processes, and they each have different properties.  It would be too much to try to summarize that entire subfield in a single answer here.