These shapefiles record the geometries of 15 intervals along geomorphically fresh lava margins as well as 4 relevant synthetic (pseudo)fractal geometries. Each eponymous shapefile records those intervals from one region (in an appropriate coordinate system): Hawaii, Iceland, Idaho, and synthetic. The margin geometries were collected in the field using differential GNSS techniques with a nominal precision of <3 cm. While walking each margin, the rover mast was manually kept vertical. Accidental tilt of the mast is the major source of error, and we estimate the data to therefore have a mean relative precision of 15 cm.