  In this paper, we are interested in the generic initial ideals of
\textit{singular} projective curves with respect to the graded lexicographic
order. Let $C$ be a \textit{singular} irreducible projective curve of degree
$d\geq 5$ with the arithmetic genus $\rho_a(C)$ in $\p^r$ where $r\ge 3$. If
$M(I_C)$ is the regularity of the lexicographic generic initial ideal of $I_C$
in a polynomial ring $k[x_0,..., x_r]$ then we prove that $M(I_C)$ is
$1+\binom{d-1}{2}-\rho_a(C)$ which is obtained from the monomial $$ x_{r-3}
x_{r-1}\,^{\binom{d-1}{2}-\rho_a(C)}, $$ provided that $\dim\Tan_p(C)=2$ for
every singular point $p \in C$. This number is equal to one plus the number of
non-isomorphic points under a generic projection of $C$ into $\p^2$. %if
$\deg(C)=3,4$ then $M(I_C)= \deg(C)$ by the direct computation. Our result
generalizes the work of J. Ahn for \textit{smooth} projective curves and that
of A. Conca and J. Sidman \cite{CS} for \textit{smooth} complete intersection
curves in $\p^3$. The case of singular curves was motivated by \cite[Example
4.3]{CS} due to A. Conca and J. Sidman. We also provide some illuminating
examples of our results via calculations done with {\it Macaulay 2} and \texttt
{Singular} \cite{DGPS, GS}.
