AbstractLetTbe a trail of a graphG. Tis aspanning trail (S‐trail)ifTcontains all vertices ofG. Tis adominating trail (D‐trail)if every edge ofGis incident with at least one vertex ofT.Acircuitis 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 Williams