We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on the Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou (1998 J. Math. Anal. Appl. 225 440–60) and Dermenjian et al (1998 Commun. Partial Differ. Equ. 23 141–69). As a technical preliminary, we carry out a spectral analysis for Schrödinger-type operators in straight Dirichlet tubes. We also apply the result to the strips embedded in abstract surfaces.