We introduce vanishing generalized Morrey spaces \documentclass[12pt]{minimal}
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\begin{document}$${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$$\end{document} with a general function \documentclass[12pt]{minimal}
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\begin{document}$${\varphi(x, r)}$$\end{document} defining the Morrey-type norm. Here \documentclass[12pt]{minimal}
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\begin{document}$${\Pi \subseteq \Omega}$$\end{document} is an arbitrary subset in Ω including the extremal cases \documentclass[12pt]{minimal}
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\begin{document}$${\Pi = \{x_0\}, x_0 \in \Omega}$$\end{document} and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces \documentclass[12pt]{minimal}
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\begin{document}$${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$$\end{document} we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type \documentclass[12pt]{minimal}
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\begin{document}$${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$$\end{document} -theorem for the potential operator Iα. The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \documentclass[12pt]{minimal}
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\begin{document}$${\varphi(x, r)}$$\end{document}. No monotonicity type condition is imposed on \documentclass[12pt]{minimal}
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\begin{document}$${\varphi(x, r)}$$\end{document}. In case \documentclass[12pt]{minimal}
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\begin{document}$${\varphi}$$\end{document} has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function \documentclass[12pt]{minimal}
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\begin{document}$${\varphi}$$\end{document}. The proofs are based on pointwise estimates of the modulars defining the vanishing spaces
