Hyperbolic metamaterials comprised of an array of plasmonic nanorods provide a unique platform for designing optical sensors and integrating nonlinear and active nanophotonic functionalities. In this work, the waveguiding properties and mode structure of planar anisotropic metamaterial waveguides are characterized experimentally and theoretically. While ordinary modes are the typical guided modes of the highly anisotropic waveguides, extraordinary modes, below the effective plasma frequency, exist in a hyperbolic metamaterial slab in the form of bulk plasmon-polaritons, in analogy to planar-cavity exciton-polaritons in semiconductors. They may have very low or negative group velocity with high effective refractive indices (up to 10) and have an unusual cut-off from the high-frequency side, providing deep-subwavelength (λ<inf>0</inf>/6-λ<inf>0</inf>/8 waveguide thickness) single-mode guiding. These properties, dictated by the hyperbolic anisotropy of the metamaterial, may be tuned by altering the geometrical parameters of the nanorod composite.