  In Hele-Shaw flows, boundaries between fluids develop unstable viscous
fingers. At vanishing surface tension, the fingers further evolve to cusp-like
singularities. We show that the problem admits a {\it weak solution} where
shock fronts triggered by a singularity propagate together with a fluid. Shocks
form a growing, branching tree of a mass deficit, and a line distribution of
vorticity where pressure and velocity of the fluid have finite discontinuities.
Imposing that the flow remain curl-free at macroscale determines the shock
graph structure. We present a self-similar solution describing shocks emerging
from a generic (2,3)-cusp singularity -- an elementary branching event.
