  In this paper, we classify the 1-Weierstrass points of the Kuribayashi
quartic curves with three parameters $a,\,b$ and $c$ defined by the equation \[
C_{a,b,c}:x^{4}+y^{4}+z^{4}+ax^{2}y^{2}+bx^{2}z^2+cy^{2}z^{2}=0, \] such that
$(a^2-1)(b^2-1)(c^2-1)(a^2-4)(b^2-4)(c^2-4)(a^2+b^2+c^2-abc-4)\neq0.$
Furthermore, the geometry of these points is investigated.
