  Consider a sequence of linear contractions $S_{j}(x)=\varrho x+d_{j}$ and
probabilities $p_{j}>0$ with $\sum p_{j}=1$. We are interested in the
self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of finite type. In
this paper we study the multi-fractal analysis of such measures, extending the
theory to measures arising from non-regular probabilities and whose support is
not necessarily an interval.
  Under some mild technical assumptions, we prove that there exists a subset of
supp$\mu $ of full $\mu $ and Hausdorff measure, called the truly essential
class, for which the set of (upper or lower) local dimensions is a closed
interval. Within the truly essential class we show that there exists a point
with local dimension exactly equal to the dimension of the support. We give an
example where the set of local dimensions is a two element set, with all the
elements of the truly essential class giving the same local dimension. We give
general criteria for these measures to be absolutely continuous with respect to
the associated Hausdorff measure of their support and we show that the
dimension of the support can be computed using only information about the
essential class.
  To conclude, we present a detailed study of three examples. First, we show
that the set of local dimensions of the biased Bernoulli convolution with
contraction ratio the inverse of a simple Pisot number always admits an
isolated point. We give a precise description of the essential class of a
generalized Cantor set of finite type. Lastly, we study a maximal loop class
that is not truly essential.
