AbstractGeometric regularity of spider webs has been intensively studied in orb‐weaving spiders, although it is not exclusive of orb weavers. Here, we document the geometrically regular, repetitive elements in the webs of the non‐orb‐weaving groups Leptonetidae and Telemidae for the first time. Similar to orb weavers, we found areas with regularly spaced parallel lines in the webs of Calileptoneta helferi, Sulcia sp., and cf. Pinelema sp. Furthermore, we provide a detailed account of the regular webs of Ochyrocera (Ochyroceratidae). The sections of the web with regularly disposed parallel lines are built as U‐shaped modules reminiscent of orb webs. It has been suggested that the regularly spaced parallel lines in the webs of Ochyroceratidae and Psilodercidae may be produced in a single sweep of their posterior lateral spinnerets, which have regularly spaced aciniform gland spigots, perhaps involving expansion of the spinnerets. To test this hypothesis, we compared the spacing between parallel lines with the spacing between spigots, searched for expansible membranes in the spinnerets, and examined the junctions of regularly spaced lines. The distance between parallel lines was 10–20 times the distance between spigots, and we found no expansible membranes, and the intersection of parallel lines are cemented, which opposes the single sweep hypothesis. Furthermore, we found cues of viscid silk in the parallel lines of the psilodercid Althepus and broadened piriform gland spigots that may be responsible of its production. Finally, we evaluated the presence or absence of geometrically regular web elements across the spider tree of life. We found reports of regular webs in 31 spider families, including 20 families that are not orb weavers and hypothesize that the two basic aspects of regularity (parallel lines spaced at regular intervals, and radial lines spaced at regular angles) probably appeared many times in the evolution of spiders.