Let \(f\in \mathbb{C} [x_1, x_2, \dots, x_n]\) and fix an embedded resolution \(h: X\to \mathbb{C}^n\) of \(f^{-1} (0)\) such that \(h^{-1} (f^{-1} (0))\) is a normal crossing divisor. We denote by \(E_i\), \(i\in T\), the reduced irreducible components of \(h^{-1} (f^{-1} (0))\), and by \(N_i\) and \(\nu_i -1\) the multiplicities of \(E_i\) in the divisor of \(f\circ h\) and \(h* (dx_1 \wedge dx_2 \wedge \cdots \wedge dx_n)\) on \(X\), respectively. For \(I\subset T\) denote also \(E_I= \bigcap_{i\in I} E_i\) and \(\overset \circ E_I= E_I\setminus (\bigcup_{j\not\in I} E_j)\). The rational function \[ Z_{\text{top}, 0} (s)= \sum_{I\subset T} \chi(E_I \cap h^{-1} (0)) \prod_{i\in I} {1\over {\nu_i+ sN_i}} \] is called the topological zeta function of the germ of \(f\) at 0, where \(\chi(\;)\) denotes the Euler-Poincaré characteristic. It does not depend on the chosen resolution. In this article, the author determines all poles of \(Z_{\text{top}, 0} (s)\) for \(n=2\) and for any \(f\in \mathbb{C} [x_1, x_2]\). Assume that \((X, h)\) is the canonical embedded resolution of \(f\), i.e., the minimum one in the set of all birational morphisms \(h: X\to \mathbb{C}^2\) such that \(h^{-1} (f^{-1} (0))\) is a normal crossing divisor with smooth components. The main results are the following two theorems: Theorem 1. \(Z_{\text{top}, 0} (s)\) has at most one pole of order 2. Moreover \(s_0\) is a pole of order 2 if and only if there exist two intersection components \(E_i\) and \(E_j\) with \(s_0= -{\nu_i \over N_i}=- {\nu_j \over N_j}\), and in that case \(s_0\) is the pole closest to the origin. Theorem 2. A complex number \(s_0\) is a pole of \(Z_{\text{top}, 0} (s)\) if and only if \(s_0=- {\nu_i \over N_i}\) for some exceptional curve \(E_i\) intersecting at least three times other components or \(s_0=- {1\over N_i}\) for some irreducible component \(E_i\) of the strict transform of \(f^{-1} (0)\). The verification relies on consideration of the resolution graph and the numbers \({\nu_i \over N_i}\).