The authors introduce a new fine classification of singularly continuous probability measures on \(\mathbb R\) on the basis of topological and metric properties of the support of a measure and the local behaviour of a measure on subsets of the support. They prove a decomposition of any singularly continuous measure into a convex combination of three measures of pure spectral types. In order to construct explicit examples of measures of the above types, the authors introduce a \(\tilde{Q}\)-representation of real numbers, an extension of the \(Q\)-representation used to construct new fractals in their earlier works (see, for example, [G. Torbin, Theory Stoch. Process. 13, No. 29, Part 1–2, 281–293 (2007; Zbl 1142.60032)]). This technique is applied also to study properties of generalized infinite Bernoulli convolutions.