  We consider selfadjoint operators obtained by pasting a finite number of
boundary relations with one-dimensional boundary space. A typical example of
such an operator is the Schr\"odinger operator on a star-graph with a finite
number of finite or infinite edges and an interface condition at the common
vertex. A wide class of "selfadjoint" interface conditions, subject to a
assumption which is generically satisfied, is considered. We investigate
properties of spectral multiplicity of singular spectrum (continuous as well as
point) in terms of the spectral data of decoupled operators.
