Several types of lemon-shaped billiards and some symmetrical billiards with discontinuities in the curvature radius are investigated classically and quantally. Special attention is paid to orbits in which the particle is reflected from the boundary at the singular points. The orbit stability is investigated in dependence on the shape parameter and compared for different boundary types. Further analysis of the presented classical billiards include the Poincare surface sections. It can be observed that for certain boundary shapes and for some ranges of the shape parameter the chaotic part of the phase plane separates into several distinct chaotic segments. In the quantal calculation the chaotic fraction of the phase space is computed. The obtained classical values are compared with the corresponding results obtained from the statistical analysis of the energy level densities. The method used is the Nearest Neighbour Spacing Distribution (NNSD), and in fitting the calculated histograms the Brody, Berry-Robnik and Berry-Robnik-Brody distributions are used. Some properties of the calculated wave functions will also be discussed.