Siegel's theorem and singular curves

Question

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the requirement is that the polynomial $F(x,y)$ defining the curve be irreducible. Is either of these really necessary? If smoothness is a requirement, how could one determine whether an irreducible, singular curve of genus 2 (for example) passes through infinitely many integral points or not?