Non-spectrality of some piecewise smooth curves and unions of line segments

We develop a systematic study about the spectrality of measures supported on piecewise smooth curves by studying the support of the tempered distributions arising from the tiling equation of some singular spectral measures. In doing so, we show that the arc-length measures of all closed polygonal lines are not spectral. {In particular, the boundary of a square is not spectral. We also show that the ``plus space'' (two crossing line segments) is not spectral.} Furthermore, our theory also shows that the arc length measures on {smooth} convex curves with finitely many transverse self-intersections are not spectral. Finally, several natural open questions about the spectrality of singular measures and {piecewise} smooth curves will also be discussed.