Although microlocal analysis was developed originally exclusively for linear problems, it has played an increasingly important role in nonlinear PDE via what's known as paradifferential calculus. There are some nice write-ups about this. Here are a few:
https://www.math.ucla.edu/~tao/247b.1.07w/notes6.pdf
http://www.ams.org/notices/201007/rtx100700858p.pdf
http://www.unc.edu/math/Faculty/met/micro2.pdf (Wayback Machine)
