Let \(X\) and \(Y\) be open sets in \(\mathbb{R}^n\) and \(\mathbb{R}^m\), respectively, and let \({\mathcal E}^*(X)\) denote the class of ultradifferentiable funcions (of Beurling or Roumieu type) associated to an appropriate sequence of positive numbers \((M_p).\) The authors study the existence of the composition \(u\circ f\) in the case that \(f = (f_1, \dots, f_m):X \to Y\), \(f_i\in {\mathcal E}^*(X),\) and the singular spectrum of \(u\in {\mathcal D}^{'*}(Y)\) does not intersect the set of normals of \(f.\) This is an extension of Theorem 8.2.4 in \textit{L. Hörmander} [``The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis'' (Grundlehren 256, Springer, Berlin etc.) (1983; Zbl 0521.35001) (\({}^2\)1990; Zbl 0712.35001) (Reprint 1983; Zbl 1028.35001)]. The proof is based on the theory of almost analytic extensions and boundary values of holomorphic functions. A corresponding result for microfunctions is also given.