ABSTRACT The 1.33-mm survey of protoplanetary discs in the Taurus molecular cloud found annular gaps and rings to be common in extended sources (≳ 55au), when their 1D visibility distributions were fit parametrically. We first demonstrate the advantages and limitations of non-parametric visibility fits for data at the survey’s 0.12-arcsec resolution. Then we use the non-parametric model in Frankenstein (frank) to identify new substructure in three compact and seven extended sources. Among the new features, we identify three trends: a higher occurrence rate of substructure in the survey’s compact discs than previously seen, underresolved (potentially azimuthally asymmetric) substructure in the innermost disc of extended sources, and a ‘shoulder’ on the trailing edge of a ring in discs with strong depletion at small radii. Noting the shoulder morphology is present in multiple discs observed at higher resolution, we postulate it is tracing a common physical mechanism. We further demonstrate how a superresolution frank brightness profile is useful in motivating an accurate parametric model, using the highly structured source DL Tau in which frank finds two new rings. Finally, we show that sparse (u, v) plane sampling may be masking the presence of substructure in several additional compact survey sources.