The family of Jacobi quartics [math] y^2 = (1-x^2) (1-e^2 x^2) [/math] typically shows no singular point in the affine plane.
Goes with my "A primer of elliptic curves" talk, slides [url=https://dev.isen.fr/courses/ComputerScience/fr/slides.html?presentation=mathematiques/misc/ec]here[/url].