A Survey of Ensemble Physical Properties of Cometary Nuclei (SEPPCoN) is underway to characterize the nuclei of 100 Jupiter-family comets (JFC). The survey combines both visible and mid-infrared observations to measure the JFC size and albedo distributions. We inspected Spitzer Space Telescope MIPS and IRS images of the survey targets for dust comae, tails, and trails. Out of 98 observed comets, we found 32 to have some emission from dust outside of the central point source. A few of these sources were also observed to have dust in visible, ground-based data. The heliocentric distances (rh) of the 32 targets range from 3.5 to 6.5 AU, with most between 4 and 5 AU. We derive color-temperatures for the 20 dust detections observed in both the 16 and 22 micron IRS cameras and find the color-temperature approximately varies as 280*rh^(-0.5) [K], as expected for isothermal low-albedo dust in local thermodynamic equilibrium. We discuss the evidence for outliers from this trend. We compare our observations to dust syndynes and 3-dimensional dust models to distinguish dust trails from dust tails. Unlike dust tails, dust trails only weakly respond to solar radiation pressure and, therefore, likely represent the largest (> 1 mm) grains ejected from the nucleus. We also compare observations to model images in order to determine the extent of recent coma activity. Water sublimation is expected to be greatly extinguished on comet surfaces by 3.5 AU. Dust structures observed outside of this rh could arise from recent coma activity (timescales up to weeks) caused by the sublimation of highly volatile ices (such as CO2) or the crystallization of amorphous water ice. Alternatively, the observed dust may be slowly dispersing grains ejected at a much earlier epoch (timescales up to years) when water sublimation dominated coma activity.