  In this paper we show that certain trigonometric polynomial shearlets which
are special cases of directional de la Vall\'{e}e Poussin type wavelets are
able to detect singularities along boundary curves of periodic characteristic
functions. Motivated by recent results for discrete shearlets in two
dimensions, we provide lower and upper estimates for the magnitude of
corresponding inner products. In the proof we use localization properties of
trigonometric polynomial shearlets in the time and frequency domain and, among
other things, bounds for certain Fresnel integrals. Moreover, we give numerical
examples which underline the theoretical results.
