The non-trivial behaviour of phase is crucial for many important physical phenomena, such as, for example, the Aharonov-Bohm effect(1) and the Berry phase(2). By manipulating the phase of light one can create 'twisted' photons(3,4), vortex knots(5) and dislocations(6) which has led to the emergence of the field of singular optics relying on abrupt phase changes(7). Here we demonstrate the feasibility of singular visible-light nano-optics which exploits the benefits of both plasmonic field enhancement and the peculiarities of the phase of light. We show that properly designed plasmonic metamaterials exhibit topologically protected zero reflection yielding to sharp phase changes nearby, which can be employed to radically improve the sensitivity of detectors based on plasmon resonances. By using reversible hydrogenation of graphene(8) and binding of streptavidin-biotin(9), we demonstrate an areal mass sensitivity at a level of fg mm(-2) and detection of individual biomolecules, respectively. Our proof-of-concept results offer a route towards simple and scalable single-molecule label-free biosensing technologies.