Singular optics are commonly explored through scalar or vectorial singularities, whereas matrix-level singularities remain far less developed for wave manipulations. Here, we experimentally demonstrate polarization-addressable singular Jones metasurfaces using reflective plasmonic nanostructures. By engineering polarization conversion in the reflection subspace, the Jones matrix can be driven to a singular condition, mapping an incident polarization eigenstate to a vectorial zero output. The corresponding eigen-polarization can be addressed across the Poincaré sphere by tailoring meta-atom geometries. Encircling the singularity in a two-dimensional parameter space can yield a full 2π topological phase winding, enabling dual-channel meta-holography and vortex generation. Extreme coupling breaking further produces circularly polarized zero-output eigenstates, revealing chiral Jones singularities. These results establish singular Jones metasurfaces as a compact platform for matrix-level singular optics, polarization-selective extinction, topological wavefront shaping, and chiral light-matter interactions.