In this paper, we are concerned with the computation of the $p$-rank and $a$-number of singular curves and their smooth model. We consider a pair $X, X'$ of proper curves over an algebraically closed field $k$ of characteristic $p$, where $X'$ is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces $S$ (with $p_g(S) = 0 = q(S)$) and $X$ is the smooth model of $X'$. We determine the $p$-rank of $X$ by using the exact sequence of group schemes relating the Jacobians $J_X$ and $J_{X'}$.