The author considers the second-order boundary value problem \[ u'' + f(t,u) = 0, \quad u(0) = u(1) = 0, \] where \(f : (0,1) \times \mathbb{R}^k \to \mathbb{R}^k\) is singular at both end points. In addition, \(f\) is either a Carathéodory or a continuous function. The operator \[ Tu(t) = \int_0^1 G(t,s) f(s,u(s))\, ds \] is defined on \(C([0,1],\mathbb{R}^k)\), where \(G(t,s)\) is Green's function of the operator \(-u''\) subject to \(u(0) =u(1) =0\). In the special case of a continuous inhomogeneous term, the following result is obtained: If \[ X_{\alpha} = \{u \in C([0,1],\mathbb{R}^k) : \sup_{t \in (0,1)} \frac{| u(t)| }{(t(1-t))^{\alpha}} 0\), \(\gamma \geq 2\), \(p > \gamma -1\) such that for all \(t \in (0,\delta) \cup (1-\delta,1)\), and \(| u| \leq \delta\), \[ | f(t,u)| \leq \frac{c| u(t)| ^p}{(t(1-t))^{\gamma}}, \] the operator \(T : X_{\alpha} \to X_{\alpha}\) is completely continuous provided \(\alpha \in (0,1)\) satisfies \(\alpha p+1 > \gamma\). The next result is based on an application of Schauder's fixed-point theorem and provides a solution under the assumption of sublinearity of \(f\): If (i) there exist \(0 0\), \(\gamma \geq 2\), \(p > \gamma -1\) such that for all \(t \in (0,\delta) \cup (1-\delta,1)\), and \(u \in \mathbb{R}^k\), \[ | f(t,u)| \leq \frac{c| u(t)| ^p}{(t(1-t))^{\gamma}}, \] and (ii) there exist \(E>0\), \(c_1>0\), \(\rho \in (0,1)\), \(\nu \leq \frac{\gamma}{p-\rho+1}\) such that for all \(t \in (0,1)\), and \(| u| \geq E\), we have \[ | f(t,u)| \leq \frac{c_1| u(t)| ^p}{(t(1-t))^{\gamma}}, \] then, for \(\alpha \in (0,1)\) with \(\alpha p+1 > \gamma\), the boundary value problem has a solution \(u \in X_{\alpha}\) with \(\| u\| \leq R\), where \[ R \geq cR^p \left(\int_0^{\delta} + \int_{1-\delta}^1\right) (s(1-s))^{\omega} \,ds + \max\{M\lambda_1, c_1\lambda_2 R^{\rho}\}, \] with \(\omega = \alpha p +1 -\alpha -\gamma\), \(\xi = \alpha \rho +1 -\alpha -\nu\), \[ \lambda_1 = \int_{\delta}^{1-\delta} (s(1-s))^{1-\alpha} \,ds, \quad \lambda_2 = \int_{\delta}^{1-\delta} (s(1-s))^{\xi} \,ds, \] and \(M = \sup\{| f(t,u)|: | u| \leq E, \delta \leq t \leq 1-\delta \}.\) Another result is based on an application of the topological degree theory: Under the assumption (H) and if \(\alpha\) satisfies \(\alpha p+1 > \gamma\), the boundary value problem has a solution in \(X_{\alpha}\) provided there exists \(M >0\) such that \((f(t,u),u) \leq 0\), \(t \in (0,1)\), for any \(| u| >M(t(1-t))^{\alpha}\). The measure \(\mu\) on \([0,1]\) is introduced by \[ \mu(A) = \int_A (s(1-s))^{1-\alpha} \,ds \] for which the space \(L_{1-\alpha}\) is defined by \[ L_{1-\alpha} = \biggl\{h : \int_0^1 (s(1-s))^{1-\alpha} | h(s)| \,ds \biggr\}, \quad \alpha \in (0,1). \] The following result is derived under the assumption that \(f\) is a Carathéodory function with respect to \(L_{1-\alpha}\): If \(f\) is a Carathéodory function (in the above sense) and if for every \(M>0\), there is \(h_M \in L_{1-\alpha}\) such that, for any \(| u| \leq M(t(1-t))^{\alpha}\), we have \(| f(t,u)| \leq h_M(t)\) a.e. \(t \in (0,1)\), then the operator \(T : X_{\alpha} \to X_{\alpha}\) is completely continuous. In this case, the author states a result similar to the existence theorem above. In addition, positive solutions are discussed in this paper. The existence theorems are illustrated by several examples.