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We consider singular $\mathbb{Q}$-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is ${ℂ}^1$- or ${ℂ}^{*}$-ruled or the surface is up to isomorphism one of two exceptional surfaces of Kodaira dimension zero. For both exceptional surfaces the Kodaira dimension of the smooth locus is zero and the singular locus consists of a unique point of type $A_1$ and $A_2$ respectively.