AbstractLet T be a trail of a graph G. T is a spanning trail (S‐trail) if T contains all vertices of G. T is a dominating trail (D‐trail) if every edge of G is incident with at least one vertex of T. A circuit is a nontrivial closed trail. Sufficient conditions involving lower bounds on the degree‐sum of vertices or edges are derived for graphs to have an S‐trail, S‐circuit, D‐trail, or D‐circuit. Thereby a result of Brualdi and Shanny and one mentioned by Lesniak‐Foster and Williamson are improved.