To study the distribution of star formation and dust emission within nearby galaxies, we measured five morphological parameters in the 3.6- and 24-μm wavebands for 65 galaxies in the Spitzer Infrared Nearby Galaxies Survey (SINGS) and eight galaxies that were serendipitously observed by SINGS. The morphological parameters demonstrate strong variations along the Hubble sequence, including statistically significant differences between S0/a-Sab and Sc-Sd galaxies. Early-type galaxies are generally found to be compact, centralized, symmetric sources in the 24-μm band, while late-type galaxies are generally found to be extended, asymmetric sources. These results suggest that the processes that increase the real or apparent sizes of galaxies' bulges also lead to more centralized 24-μm dust emission. Several phenomena, such as strong nuclear star formation, Seyfert activity, or outer ring structures, may cause galaxies to deviate from the general morphological trends observed at 24 μm. We also note that the 24-μm morphologies of Sdm-Im galaxies are quite varied, with some objects appearing very compact and symmetric but others appearing diffuse and asymmetric. These variations reflect the wide variation in star formation in irregular galaxies as observed at other wavelengths. The variations in the 24-μm morphological parameters across the Hubble sequence mirror many of the morphological trends seen in other tracers of the ISM and in stellar emission. However, the 24-μm morphological parameters for the galaxies in this sample do not match the morphological parameters measured in the stellar wavebands. This implies that the distribution of dust emission is related to but not equivalent to the distribution of stellar emission.