The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear example will be the stationary Kolmogorov equation $y u_x - u_{yy}=f$, which we investigate in a rectangle $(x_0,x_1)\times(-1,1)$, supplemented with boundary conditions on the ``parabolic boundary'' of the domain: the top and lower boundaries $\{y=\pm 1\}$, and the lateral boundaries $\{x_0\}\times (0,1)$ and $\{x_1\}\times (-1,0)$. We first prove that this equation admits a finite number of singular solutions associated with regular data. These singular solutions, of which we provide an explicit construction, are localized in the vicinity of the points $(x_0,0)$ and $(x_1,0)$. Hence, the solutions to the Kolmogorov equation associated with a smooth source term $f$ are regular if and only if $f$ satisfies a finite number of orthogonality conditions. This is similar to well-known phenomena in elliptic problems in polygonal domains. We then extend this theory to the Vlasov--Poisson--Fokker--Planck system $y u_x + E[u] u_y - u_{yy}=f$, and to two quasilinear equations: the Burgers type equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, and the Prandtl system in the vicinity of a recirculating solution, close to the curve where the horizontal velocity changes sign. We therefore revisit part of a recent work by Iyer and Masmoudi. For the two latter quasilinear equations, we introduce a geometric change of variables which simplifies the analysis. In these new variables, the linear differential operator is very close to the Kolmogorov operator $y \partial_x - \partial_{yy}$. Stepping on the linear theory, we prove existence and uniqueness of regular solutions for data within a manifold of finite codimension, corresponding to some nonlinear orthogonality conditions. Treating these three nonlinear problems in a unified way also allows us to compare their structures. In particular, we show that the vorticity formulation of the Prandtl system, in an adequate set of variables, is very similar to the Burgers one. As a consequence, solutions of the Prandtl system are actually smoother than the ones of Burgers, which allows us to have a theory of weak solutions of the Prandtl system close to the recirculation zone.