Let f: (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a germ of an analytic function with isolated singularity. Then there exists a Morsification \(f_ s: {\mathbb{C}}^{n+1}\to {\mathbb{C}}\), which has only \(A_ 1\)- singularities. The author generalizes this fact to the case in which the singular locus \(\Sigma\) of \(f^{-1}(0)\) is a curve. Under this new circumstance, one needs to add \(A_{\infty}\) and \(D_{\infty}\) singularities to the list of the allowable singularities of \(f_ s\). Here \(A_{\infty}\) and \(D_{\infty}\) are respectively defined by \(y^ 2_ 1+...+y^ 2_ n=0\) and \(y_ 0y^ 2_ 1+...+y^ 2_ n=0\) for some local coordinates \((y_ 0,y_ 1,...,y_ o)\). The main theorem states that if \(\Sigma\), defined by the ideal I, is smoothable and if \(f\in I^ 2\), then f has a Morsification. The upper bound for the number of \(D_{\infty}\) singularities is also investigated.