The long-sought quantum spin liquid (QSL) is an exciting class of magnetic spin system that remains in a dynamic, entangled state down to extremely low temperatures and is expected to exhibit novel fractional excitations such as spinons. While they have been treated theoretically for decades, it is only in the last few years that several promising QSL candidates have been discovered experimentally on various geometrically frustrated lattices. I will discuss the application of local probes, nuclear magnetic resonance (NMR) and muon spin relaxation (muSR), to such systems, with a focus on two particular materials. First, the near-kagome system Vesignieite, initially proposed to have a liquid ground state, is found to exhibit a transition to an unusual heterogeneous ground state that consists of weak static magnetism and a slowly fluctuating liquid component. Second, I will discuss a new spin-1/2 triangular lattice antiferromagnet Ba3CuSb2O9 which shows no spin freezing down to 20 mK in zero field, making it a good QSL candidate. In-field NMR experiments show a vanishing susceptibility and an exponentially slowing spin relaxation, implying gapped excitations. I will discuss several common themes observed within the small set of existing QSL candidates and the outlook for future studies of such materials.