The spin quantum Hall transition (or class C transition in two dimensions) represents one of the few localization-delocalization transitions for which some of the critical exponents are known exactly. Not known, however, is the multifractal spectrum τq, which describes the system-size scaling of inverse participation ratios Pq, i.e., the q moments of critical wave-function amplitudes. We here report simulations based on the class C Chalker-Coddington network and demonstrate that τq is (essentially) a quartic polynomial in q. Analytical results fix all prefactors except the quartic curvature that we obtain as γ=(2.22±0.15)×10−3. In order to achieve the necessary accuracy in the presence of sizable corrections to scaling, we have analyzed the evolution with system size of the entire Pq-distribution function. As it turns out, in a sizable window of q values this distribution function exhibits a (single-parameter) scaling collapse already in the preasymptotic regime, where finite-size corrections are not negligible. This observation motivates us to propose new, original approach for extracting τq based on concepts borrowed from the Kolmogorov-Smirnov test of mathematical statistics. We believe that our work provides the conceptual means for high-precision investigations of multifractal spectra also near other localization-delocalization transitions of current interest, especially the integer (class A) quantum Hall effect.