I can give a brief description of a (perhaps, currently the main) topic pertaining the application of microlocal analysis to the study of nonlinear PDEs, which seem the argument you are most interested to, since it is also the one to which I am more accustomed having been interested in it many years ago. To my knowledge, the main (if not the only) method used to extend this analysis to nonlinear PDEs, is the use of paradifferential operators and paraproducts (introduced by Jean-Michel Bony), so let's sketch how this approach works.
Notation
I am taking the monograph [2] as a reference, and assume as known (at least formally) the basic linear theory of pseudodifferential operators calculus. As it is customary when dealing with $\psi$DOs, we have
$$
\begin{split}
D_k&\overset{\text{Def}}{=}\frac{1}{i}\frac{\partial}{\partial x_k}=\frac{1}{i}\partial_{x_k}\quad k=1,\ldots,n\\
D&\overset{\text{Def}}{=}\frac{1}{i}\left(\frac{\partial}{\partial x_1},\cdots,\frac{\partial}{\partial x_n}\right)\\
D^\beta&\overset{\text{Def}}{=}\left(\frac{1}{i}\frac{\partial}{\partial x_1}\right)^{\beta_1}\!\!\!\!\cdot\ldots\cdot\left(\frac{1}{i}\frac{\partial}{\partial x_n}\right)^{\beta_n}\quad\beta=(\beta_1,\ldots,\beta_n)\in\Bbb N^n\\
\end{split}
$$
Introduction
Microlocal analysis is, roughly speaking, the analysis of the wavefront set ${W\!F}(u)$ of a distribution $u\in \mathscr{D}^\prime(\Omega)$, $\Omega\subseteq\Bbb R^n$ which solves, in the standard case, a given linear PDE (or $\psi$DE). This is generically achieved with the use of the instruments of pseudodifferental calculus (parametrices, composition operators and commutator formulas) and a priori estimates. Applying this machinery to nonlinear problems implies being able to represent nonlinear operators in terms of standard pseudodifferential operators: this is the approach taken by Bony, Meyer and their pupils.
Paradifferential operators and paraproducts
Consider a $C^\infty$ nonlinear function $F:\Bbb R\to\Bbb R$ (just to focus on the basic ideas: the method is well suited also for vector functions), a function $u\in C^r(\Bbb R^n)$ and a dyadic partition of unity of $\Bbb R^n$, $\{\psi_j(\xi)\}_{j\in \Bbb N}$ (see Taylor [2], §1.3 pp. 41-42, for the details): then we can define a countable family of constant coefficients pseudodifferential operators $\{\psi_j(D)\}_{j\in \Bbb N}$ and the associated family $\{\Psi_j(D)\}_{j\in \Bbb N}$ as follows
$$
\Psi_j(D)=\sum_{i\le j}\psi_i(D)\quad j\in \Bbb N.
$$
Applying it to $u$ we obtain
$$
u_j=\Psi_j(D)u\implies u(x)=\sum_{j=0}^\infty \psi_j(D)u(x)=\lim_{j\to\infty}u_j(x)
$$
and consequently
$$
F(u)=F(u_0)+\big[F(u_1)-F(u_0)\big]+\big[F(u_2)-F(u_1)\big]+\cdots+\big[F(u_{k+1})-F(u_k)\big]+\cdots
$$
Now we have that
$$
\begin{split}
F(u_{k+1})-F(u_k)&= F\big(u_k+\psi_{k+1}(D)u\big)-F(u_k)\\
&=m_k(x)\psi_k(D)u
\end{split}\quad\forall k\in\Bbb N
$$
where
$$
m_k(x)=\int_0^1\!\!\! F^\prime\big(u_k+t\psi_{k+1}(D)u\big)\mathrm{d}t =
\int_0^1\!\!\! F^\prime\big(\Psi_j(D)u+t\psi_{k+1}(D)u\big)\mathrm{d}t\quad\forall k\in\Bbb N
$$
Thus we have obtained the following decomposition
$$
F(u) = F(u_0)+\sum_{k=1}^\infty m_k(x)\psi_k(D)u = M_F(x,D)u + R(u)\label{1}\tag{1}
$$
where
- $M_F(x,D)u\overset{\text{Def}}{=} \sum_{k=1}^\infty m_k(x)\psi_k(D)u$ is a paradifferential operator, which is a particular kind of $\psi$DO by construction.
- $R(u)=F(u_0)$ is a remainder term, which is $C^\infty$-smooth by construction, since each $\psi_j(\xi)$, $j\in \Bbb N$, is a compactly supported $C^\infty$-function.
We have thus obtained a general nonlinear function composition as the sum of a (infinitely) smooth remainder term $R$ and of a paradifferential operator: now, considering a nonlinear partial differential equation
$$
F(x,u,\ldots,\partial^\beta u,\ldots)_{|\beta|\le m}=f\quad m\in\Bbb N\label{nlpde}\tag{NLPDE}
$$
we can proceed in a similar way as for \eqref{1} and obtain the following formula
$$
\begin{split}
F(x,u,\ldots,\partial^\beta u,\ldots)_{|\beta|\le m}&=\sum_{|\beta|\le m} M_{\partial_{\partial^\beta u}F}(x,D)\partial^\beta u + F(x,u_0,\ldots,\partial^\beta u_0,\ldots)\\
&=\sum_{|\beta|\le m} M_{\partial_{\partial^\beta u}F}(x,D)\partial^\beta u + R(x,u_0,\ldots,\partial^\beta u_0,\ldots)
\end{split}\label{2}\tag{2}
$$
Thus again we have obtained an expansion expressing a nonlinear PDE as the sum of a finite number of paradifferential operators and a $C^\infty$-smooth remainder term: then you can apply to formula \eqref{2} the machinery of microlocal analysis developed for the linear theory and get a priori estimates and information on the structure of the wavefront set $W\!F(u)$.
The paraproduct is instead a linear operator related to the operator $M_F(x,D)$ in formula \eqref{1}. There are several versions of it: one is the following
$$
\pi(a,u)=\sum_{j\ge 1}^\infty \big(\Psi_{j-1}(D)a\big)\big(\psi_{j+1}(D)u\big)
\quad a, u\in C^r\label{3}\tag{3}
$$
Apart from its elementary properties, \eqref{3} can be used in approximating $M_F(x,D)u$ and obtain a formula analogous to \eqref{1} (see [2], p. 76, proposition 3.2.C)
$$
F(u)=\pi\big(F^\prime(u),u\big)+R(u)\label{4}\tag{4}
$$
where $R$ belongs now to an appropriate Sobolev space. The analogies between this formula and \eqref{1} can be used in the study of \eqref{nlpde} to obtain from \eqref{4} a representation similar to \eqref{2} and study it by the techniques of microlocal analysis. Not being an expert in microlocal analysis nor having been involved in studying the subject recently, I address to the monographs [1] and [2] for the details of this sketchy answer.
Notes
The book [1] is a nice and well organized illustration of known results in the microlocal analysis of nonlinear PDEs, and exposes many results on the Navier-Stokes system including the ones obtained by Marco Cannone, who proved an important global existence theorem for the solution of this system of PDEs, assuming the Cauchy data belongs to an appropriate Besov space.
See also the MathOverflow question:-"Why is paraproduct or paradifferential calculus important in PDE theory? In particular, the original paper of J-M. Bony linked in the answer to that question is very clear and informative.
Bibliography
[1] Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin (2011), Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 343, Berlin-Heidelberg-Dordrecht-London-New York: Springer-Verlag, pp. XVI+524, DOI 10.1007/978-3-642-16830-7, ISBN 978-3-642-16829-1, e-ISBN 978-3-642-16830-7, MR2768550, Zbl 1227.35004.
[2] Michael Eugene Taylor (1991), Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, vol. 100. Boston, MA-Basel-Berlin: Birkhäuser Verlag, pp. 213, ISBN: 0-8176-3595-5, MR1121019, Zbl 0746.35062.
