Gonality of non-Gorenstein curves of genus five

We establish sufficient conditions for some curves to be trigonal and derive from them that most of non-Gorenstein curves of genus five are so. Afterwards, we show that the gonality of such a curve ranges from 2 to 5. Gonality is understood within a broader context, i.e., the gd1 may possibly admit a base point and correspond to a torsion free sheaf of rank one instead of a line bundle. This study comes along with a thorough description of possible canonical models and kinds of singularities.