In the range 4 < \kappa < 8, the intersection of the Schramm-Loewner Curve (one of the central objects in the theory of 2-D Conformally Invariant Systems) with the boundary of its domain is a random fractal set. After reviewing some previous results on the dimension and measure of this set, I will describe recent joint work with Ilia Binder and Fredrik Viklund that partitions this set of points according to the generalized "angle" at which the curve hits the boundary, and computes the Hausdorff dimension of each partition set. The Hausdorff dimension as a function of the angle is what we call the dimension spectrum.