The authors study the Bernoulli measure \(v_\lambda\), which is the probability distribution of the random series \(\sum_{n \geq 0} \pm \lambda^n\) for generic \(1/2 < \lambda < 1\). Specifically, they show that this measure is in the Sobolev space \(W^{s,2}\) for a.e. \(\lambda < 0.649\) such that \(\lambda^{1+2s} > 1/2\). Also, in the range \(\lambda > 1/2 + \varepsilon\), the dimension of the exceptional values of \(\lambda\) where the measure is singular is at most \(1-c\varepsilon\) for some absolute constant \(c>0\). Similar results are obtained for asymmetric Bernoulli convolutions and for variants such as {0,1,3} sets. Other results of the same flavor are obtained, e.g., for a set \(E\) in \(R^d\) of dimension at least 2, the orthogonal projection to a line has non-empty interior for all but an exceptional set of directions of dimension at most \(d+1- \dim(E)\). A similar result is obtained for the distance set between a Borel set of a certain dimension and a generic point. The results are obtained as part of a more systematic study of generic projections of a fixed large-dimensional set to a small-dimensional space under a map which depends on a certain number of parameters. Some general results for these problems are obtained.