The detailed morphology of the interstellar medium (ISM) in the central kiloparsec of galaxies is controlled by pressure and gravitation. The combination of these forces shapes both circumnuclear star formation and the growth of the central, supermassive black hole. We present visible and near-infrared Hubble Space Telescope images and color maps of 123 nearby galaxies that show the distribution of the cold ISM, as traced by dust, with excellent spatial resolution. These observations reveal that nuclear dust spirals are found in the majority of active and inactive galaxies and they possess a wide range in coherence, symmetry, and pitch angle. We have used this large sample to develop a classification system for circumnuclear dust structures. In spite of the heterogeneous nature of the complete sample, we only find symmetric, two-arm nuclear dust spirals in galaxies with large-scale bars, and these dust lanes clearly connect to dust lanes along the leading edges of the large-scale bars. Not all dust lanes along large-scale bars form two-arm spirals, however, and several instead end in nuclear rings. We find that tightly wound, or low pitch angle, nuclear dust spirals are more common in unbarred galaxies than barred galaxies. Finally, the extended narrow-line region in several of the active galaxies is well resolved. The connection between the ionized gas and circumnuclear dust lanes in four of these galaxies provides additional evidence that a significant fraction of their extended narrow-line region is ambient gas photoionized in situ by the active nucleus. In a future paper we will use our classification system for circumnuclear dust to identify differences between active and inactive galaxies, as well as barred and unbarred galaxies, in well-matched subsamples of these data.