The author shows how to explicitely calculate the rational cohomology groups of two particular moduli spaces of complex algebraic curves. More precisely, let \(M_{3,2}\) be the moduli space of non-singular complex projective curves of genus 3 with two marked points, and denote by \(Q_2\) the moduli space of plane quartic curves with two marked points. It is well known that \(Q_2\) is the complement of the hyperelliptic locus \(H_{3,2}\) inside \(M_{3,2}\), which already nourishes the expectation that the rational cohomology groups of these two moduli spaces should be somehow related. In fact, the author’s main theorem describes the explicit structure of \(H^k(Q_2,\mathbb{Q})\) and of \(H^k (M_{3,2},\mathbb{Q})\), thereby stating that these groups are simultaneously trivial for \(k=1\), \(k=3\), \(k=7\), or \(k\geq 9\). Moreover, it turns out that these groups only differ in the three particular cases of \(k=2\), \(k=4\) and \(k=8\). The method of proof is based on the idea of relating the rational cohomology of the space \(Q_2\) to that of a suitable, very concrete incidence variety \(I_2\), which is then computed as the cohomology of a fibre space via the so-called Vassiliev-Gorinov method for the cohomology of complements of discriminantal varieties [cf.: V. A. Vassiliev, Proc. Steklov Inst. Math. 225, 121–140 (1999; Zbl 0981.55008)]. The basic ingredient of this approach is a moduli-adapted version of a recent result due to C. A. M. Peters and J. H. M. Steenbrink [Mosc. Math. J. 3, No. 3, 1085–1095 (2003; Zbl 1049.14035)], which is explained in another, parallel and closely related paper by the author and J. Bergström [Math. Ann. 338, No. 1, 207–239 (2007; Zbl 1126.14030)]. It should be recalled that the Vassiliev-Gorinov method of computing the cohomology of certain discriminantal varieties is heavily based on the framework of Borel-Moore homology, which accordingly plays an equally dominant role in the explicit calculations carried out in the present paper.