The free convolution is the binary operation on the set of prob- ability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent uni- tarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free con- volution λ allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups (μt) of probability measures such that μ0 = δ0 and such that for all t> 0 and all probability measure ν, μtν (or, in the rectan- gular context, μtλν) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, for , we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semi- groups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for μ in a λ-continuous semigroup, μλν either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, and thus the expected property does not hold. However, we give suffi- cient conditions for analyticity of the density of μλν except on a negligible set of points, as well as existence and continuity of a density everywhere.