The author considers a class of mixed-type PDE of the form $Pu\equiv [D^2_y+yA(x,y,D_x)+B(x,y,D_x,D_y)]u=f$ in $\Omega\subset\bold R^{n+1}=\bold R^n_x\times\bold R_y$, where $A$ is an elliptic differential operator in $D_x$. The domain $\Omega$ intersects both the elliptic $(y>0)$ and hyperbolic $(y<0)$ half-planes. A Dirichlet boundary condition is imposed. Because the presence of a boundary value on the whole hyperbolic part of $\partial\Omega$ overdetermines this problem, singularities of the solutions must appear. The author then gives three principles for the propagation of singularities. (1) There are no isolated singularities; they cannot be trapped over compact sets in the interior of $\Omega$. (2) Any interior singularity can appear only together with singularities on the hyperbolic boundary, i.e., $\partial\Omega\cap\{y<0\}$. (3) Any singularity on the parabolic boundary $\partial\Omega\cap\{y=0\}$ will either (a) propagate into $\Omega$ when the interior normal vector at $\partial\Omega\cap\{y=0\}$ points into $y<0$ (this is the diffractive effect) or (b) remain isolated in the points of $\partial\Omega\cap\{y=0\}$ when the interior normal vector points into $y>0$, which is the gliding-ray effect.