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Let Y denote a smooth, projective curve of genus g defined over C. A point P G Y is a Weierstrass point if there exists a rational function on Y with a pole only at P of order at most g, or if there exists a regular differential on Y which vanishes at P to order at least g, or if the divisor gP is special. However, the most "functorial" way to define Weierstrass points is as the zeros of the wronskian, a section of the (g(g + l)/2)th tensor power of the canonical bundle on Y. It is this last definition that we use as the foundation for defining Weierstrass points on singular curves. What is essential is that the sheaf of dualizing differentials should be locally free and this is exactly the property satisfied by Gorenstein curves. Let X be an integral, projective Gorenstein curve of arithmetic genus g > 0 over C. Let ω denote the bundle of dualizing differentials on X and let Sf denote an invertible sheaf on X. Put s = dim H°(X,J?) = h°(&). Assume s > 0 and choose a basis φΪ9...9φsfoτH0(X,5?). We will define a section of S?®sΘω^s'1^2 as follows: Suppose that {U^} is a covering of X by open subsets such that ^f(U^) and ω(UM) are free ^(^ (α) )-modules generated by ψ(a) and τ^a\ respectively. Define FJ f e YilJ^^x) inductively