  We show that in many parametrized families of self-similar measures, their
projections, and their convolutions, the set of parameters for which the
measure fails to be absolutely continuous is very small - of co-dimension at
least one in parameter space. This complements an active line of research
concerning similar questions for dimension. Moreover, we establish some
regularity of the density outside this small exceptional set, which applies in
particular to Bernoulli convolutions; along the way, we prove some new results
about the dimensions of self-similar measures and the absolute continuity of
the convolution of two measures. As a concrete application, we obtain a very
strong version of Marstrand's projection theorem for planar self-similar sets.
