Numerous optical technologies and quantum optical devices rely on the controlled coupling of a local emitter to its photonic environment, which is governed by the local density of optical states (LDOS). Although precise knowledge of the LDOS is crucial, classical optical techniques fail to measure it in all of its frequency and spatial components. Here, we use a scanning electron beam as a point source to probe the LDOS. Through angular and spectral detection of the electron-induced light emission, we spatially and spectrally resolve the light wave vector and determine the LDOS of Bloch modes in a photonic crystal membrane at an unprecedented deep-subwavelength resolution (30-40 nm) over a large spectral range. We present a first look inside photonic crystal cavities revealing subwavelength details of the resonant modes. Our results provide direct guidelines for the optimum location of emitters to control their emission, and key fundamental insights into light-matter coupling at the nanoscale.