Context. The protoplanetary disk around the F-type star HD 135344B (SAO 206462) is in a transition stage and shows many intriguing structures both in scattered light and thermal (sub-)millimeter emission which are possibly related to planet formation processes. Aims: We aim to study the morphology and surface brightness of the disk in scattered light to gain insight into the innermost disk regions, the formation of protoplanets, planet-disk interactions traced in the surface and midplane layers, and the dust grain properties of the disk surface. Methods: We have carried out high-contrast polarimetric differential imaging (PDI) observations with VLT/SPHERE and obtained polarized scattered light images with ZIMPOL in the R and I-bands and with IRDIS in the Y and J-bands. The scattered light images and surface brightness profiles are used to study in detail structures in the disk surface and brightness variations. We have constructed a 3D radiative transfer model to support the interpretation of several detected shadow features. Results: The scattered light images reveal with unprecedented angular resolution and sensitivity the spiral arms as well as the 25 au cavity of the disk. Multiple shadow features are discovered on the outer disk with one shadow only being present during the second observation epoch. A positive surface brightness gradient is observed in the stellar irradiation corrected (r2-scaled) images in southwest direction possibly due to an azimuthally asymmetric perturbation of the temperature and/or surface density by the passing spiral arms. The disk integrated polarized flux, normalized to the stellar flux, shows a positive trend towards longer wavelengths which we attribute to large (2πa ≳ λ) aggregate dust grains in the disk surface. Part of the non-azimuthal polarization signal in the Uφ image of the J-band observation can be attributed to multiple scattering in the disk. Conclusions: The detected shadow features and their possible variability have the potential to provide insight into the structure of and processes occurring in the innermost disk regions. Possible explanations for the presence of the shadows include a 22° misaligned inner disk, a warped disk region that connects the inner disk with the outer disk, and variable or transient phenomena such as a perturbation of the inner disk or an asymmetric accretion flow. The spiral arms are best explained by one or multiple protoplanets in the exterior of the disk although no gap is detected beyond the spiral arms up to 1.''0. Based on observations collected at the European Southern Observatory, Chile, ESO No. 095.C-0273(A) and 095.C-0273(D).