We study the combinatorial geometry of a random closed multicurve on a surface of large genus g and of a random square-tiled surface of large genus g. We prove that primitive components γ₁, ⋯ , γ_k of a random multicurve m₁γ₁+ ⋯ + m_kγ_k represent linearly independent homology cycles with asymptotic probability 1 and that all its weights m_i are equal to 1 with asymptotic probability 2/2. We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus g are both very well approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of 3 g- 3 elements. In particular, we prove that the expected value of these quantities has asymptotics (log (6 g- 6) + γ) / 2 + log 2 as g→ ∞, where γ is the Euler–Mascheroni constant. These results are based on our formula for the Masur–Veech volume Vol Q_g of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform asymptotic formula for intersection numbers of ψ-classes on M¯ _g,n for large g proved by A. Aggarwal in 2020.