The author deals with the multiplicities of singularities of eigenfunctions for the Laplace-Beltrami operator. Let \(M\) be a smooth compact 2-dimensional Riemann manifold. Let \(\chi(M)\) be the Euler characteristic of the manifold \(M\). Let \(f_N\) be the eigenfunction of the Laplace-Beltrami operator on \(M\) with index \(N\). Set \(\Gamma= \{x\in M/\partial M: f_N(x)= 0\}\). Assume that all the eigenfunctions are smooth. This is the case for \(S^2\), \(\mathbb{R} P^2\), and \(T^2\). Assume that there exists a neighborhood \(U\) of the boundary such that the set \(\Gamma \cap U\) is the union of some continuous curves. Denote by \(m_\ell\) the number of points \(c\) in the boundary of the manifold \(M\) such that in a neighborhood of the point \(c\) the set \(\Gamma\) consists of \(\ell\) branches entering \(c,\ell\in [1,+ \infty)\). The main theorem in this paper is: Theorem. \(\sum_{a\in \Gamma} (k(a)- 1)+ \sum m_\ell \ell/2\leq N- \chi (M)\). Here \(k(a)\) is the degree of the principal homogeneous part of the function \(f_N\) at the point \(a\in \Gamma\). If \(M\) is \(\mathbb{R} P^2\) or \(S^2\), then a sharper estimate holds (see Theorem 2 in this paper).