The first result of this paper is that, for truncated shift operators, S- spectrality and the spectrality in the sense of Vasyunin are equivalent. Then, by applying several results of Vasyunin, it is shown that the set of the spectral singularities \(S(T_ F)\) of the truncated shift operator \(T_ F\) contains the support of the singular continuous part of the representing measure of the characteristic function F. Moreover, there are truncated shifts with arbitrary closed sets on the unit circle as sets of spectral singularities and with prescribed types of characteristic functions. The set of spectral singularities in the strict sense \(\hat S(T_ F)\) contains the support of the representing singular measure of F and even if F is a Blaschke product the sets \(S(T_ F)\) and \(\hat S(T_ F)\) can be either equal or very far from each other.