Line graphs constitute a rich and well-studied class of graphs. In this paper, we focus on three different topics related to line graphs of subcubic triangle-free graphs. First, we show that any such graph G has an independent set of size at least 3|V (G)|/10, the bound being sharp. As an immediate consequence, we have that any subcubic triangle-free graph G, with ni vertices of degree i, has a matching of size at least 3n1/20+3n2/10+9n3/20. Then we provide several approximate min-max theorems relating cycle-transversals and cyclepackings of line graphs of subcubic triangle-free graphs. This enables us to prove Jones’ Conjecture for claw-free graphs with maximum degree 4. Finally, we concentrate on the computational complexity of Feedback Vertex Set, Hamiltonian Cycle and Hamiltonian Path for subclasses of line graphs of subcubic triangle-free graphs. © 2017 Elsevier B.V. All rights reserved.