This paper is devoted to the study of different types of twisting points of conformal maps. We define the sets of gyration, spiral and oscillation points and we prove, in the case that f is conformal almost nowhere, that the above sets have Hausdorff dimension one. Also we define points of bounded radial oscillation. It is proved that there are always points of π-bounded radial oscillation but there exists a conformal map without points of small bounded radial oscillation.