Structured light is a custom light field where the phase, polarization, and intensity vary with position. It has been used for nanotweezers, nanoscale imaging, and quantum information technology, but its role in exciting optical transitions in materials has been little examined so far. Here we use group theory to derive the optical selection rules for nanosystems that get excited by structured light. If the size of the nanostructure is comparable to the light wavelength, it will sample the full beam profile during excitation with profound consequences on optical excitations. Using nano-oligomers as model nanosystems, we show that structured light excites optical transitions that are forbidden for linearly polarized or unpolarized light. Such dipole forbidden modes have longer lifetimes and narrower resonances than dipole-allowed transitions. We derive symmetry-adapted eigenmodes for nano-oligomers containing up to six monomers. Our study includes tables with selection rules for cylindrical vector beams, for beams with orbital angular momentum, and for field retardation along the propagation direction. We discuss multiphoton processes of nonlinear optics in addition to one-photon absorption. Structured light will unlock a broad range of excitations in nano-oligomers and other nanostructures that are currently inaccessible to optical studies.