I found these curve lists:
- http://www.lmfdb.org/EllipticCurve/
- https://en.wikipedia.org/wiki/List_of_curves
- http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html
- http://old.nationalcurvebank.org/home/home.htm
- http://www.2dcurves.com/
- http://www.mathcurve.com/courbes2d/courbes2d.shtml
But they aren't very structured. Wondering if there is anything with on the order of thousands of highly structured curves.
I'm assuming that by structured, you mean listed in categories of similar types. My standard references in that regard are
A Catalog of Special Plane Curves, J. Dennis Lawrence, Dover, 1972
CRC Standard Curves and Surfaces, David von Seggern, CRC Press, 1993.
And some auxiliary references
Gnomon: from Pharoahs to fractals, Michael J. Gazal$\acute{\text{e}}$, Princeton University Press, 1999
Spiral: from Theodorus to Chaos, Philip J. Davis, A.K. Peters, Ltd., 1993.