  For the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x~=~0$, it
is well known that solutions can develop singularities in finite time. For an
open dense set of initial data, the present paper provides a detailed
asymptotic description of the solution in a neighborhood of each singular
point, where $|u_x|\to\infty$. The different structure of conservative and
dissipative solutions is analyzed.
