Context. The study of dynamical processes in protoplanetary disks is essential to understand planet formation. In this context, transition disks are prime targets because they are at an advanced stage of disk clearing and may harbor direct signatures of disk evolution. Aims. We aim to derive new constraints on the structure of the transition disk MWC 758, to detect non-axisymmetric features and understand their origin. Methods. We obtained infrared polarized intensity observations of the protoplanetary disk MWC 758 with VLT/SPHERE at 1.04 µm to resolve scattered light at a smaller inner working angle (0.093 ) and a higher angular resolution (0.027 ) than previously achieved. Results. We observe polarized scattered light within 0.53 (148 au) down to the inner working angle (26 au) and detect distinct nonaxisymmetric features but no fully depleted cavity. The two small-scale spiral features that were previously detected with HiCIAO are resolved more clearly, and new features are identified, including two that are located at previously inaccessible radii close to the star. We present a model based on the spiral density wave theory with two planetary companions in circular orbits. The best model requires a high disk aspect ratio (H/r ∼ 0.20 at the planet locations) to account for the large pitch angles which implies a very warm disk. Conclusions. Our observations reveal the complex morphology of the disk MWC 758. To understand the origin of the detected features, the combination of high-resolution observations in the submillimeter with ALMA and detailed modeling is needed.