This talk deals with the regularity properties (including propagation and interaction of nonlinear waves) of the solutions of the Cauchy problem to 2D semilinear wave equation with the removable singularities of the solutions of fully nonlinear hyperbolic systems arising in the mechanics of compressible fluids with constant entropy, and with the regularizing properties of the multidimensional wave equation with dissipative term. We shall first discuss the machinery of the pseudodifferential, respectively paradifferential operators which is applied. More precisely, "radially smooth" initial data having singularities on a "massive" set of angles in the plane, including the Cantor continuum set, yield singularities propagating as in the linear case. There is a big difference between the 2D case and the multidimensional case (3D) when the interaction of several (for example four) characteristic hyper-planes could produce singularities on a dense subset of the compliment of the light cone of the future located over the origin. A result of Bony for the triple interaction of progressing linear waves in the 2D case is commented too as then new effect appears: new born wave propagating along the cone of the future with vertex at the origin. We assume that the first variation of the nonlinear system under consideration is linear, symmetric and positive one in the sense of Friedrichs. A microlocal version of the Moser's condition on the existence of global solutions on the torus of the same system enables us to prove the nonexistence of isolated singularities at each characteristic point of the main symbol of the first variation. For symmetric quasilinear hyperbolic systems we study the propagation of regularity. As usual, the strength of the singularities is measured both in Sobolev space and microlocalized Sobolev spaces. An example from fluid mechanics will be presented in order to illustrate our results.