Phonon transport is a dominant mechanism of thermal conduction in solids that has been studied for decades. A good understanding of many transport regimes in micro- and nanostructures has been established, including ballistic and diffusive transport, mode softening, or band structure engineering in phononic crystals. However, the limit of quantized transport and the engineering of single transport channels is much less explored. In this Perspective, we discuss concepts and theoretical and experimental progress in the field of quantized phonon transport in channels, such as molecular systems. We particularly highlight open questions and research opportunities that should be within experimental reach. Challenges in experimental sensitivity and control hinder fast experimental progress. Recently, however, heat transport measurements through quantum channels and single molecules have become available at room temperature using break junction techniques. These techniques are well established in the molecular electronics community and have recently been expanded to the measurement of heat transport on the single-molecule level. Given the new experimental capabilities, it is now inviting to address the rather unexplored area of molecular phonon-engineering. Several interesting theoretical predictions concern the realization of the phonon quantum interference effect, suppression of phonon current via the introduction of side groups to molecules, and the construction of a phonon diode device based on molecular anharmonicity and asymmetry. This Perspective should serve the experimental and theory community by listing key challenges, thus a roadmap for making progress in the field of quantized phonon transport.