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We prove that square-tiled surfaces having fixed combinatorics of horizontalcylinder decomposition and tiled with smaller and smaller squares becomeasymptotically equidistributed in any ambient linear $GL(\mathbb R)$-invariantsuborbifold defined over $\mathbb Q$ in the moduli space of Abeliandifferentials. Moreover, we prove that the combinatorics of the horizontal andof the vertical decompositions are asymptotically uncorrelated. As aconsequence, we prove the existence of an asymptotic distribution for thecombinatorics of a "random" interval exchange transformation with integerlengths.  We compute explicitly the absolute contribution of square-tiled surfaceshaving a single horizontal cylinder to the Masur-Veech volume of any ambientstratum of Abelian differentials. The resulting count is particularly simpleand efficient in the large genus asymptotics. We conjecture that thecorresponding relative contribution is asymptotically of the order $1/d$, where$d$ is the dimension of the stratum, and prove that this conjecture isequivalent to the long-standing conjecture on the large genus asymptotics ofthe Masur-Veech volumes. We prove, in particular, that the recent results ofChen, M\"oller and Zagier imply that the conjecture holds for the principalstratum of Abelian differentials as the genus tends to infinity.  Our result on random interval exchanges with integer lengths allows to makeempirical computation of the probability to get a $1$-cylinder pillowcase covertaking a "random" one in a given stratum. We use this technique to derive theapproximate values of the Masur-Veech volumes of strata of quadraticdifferentials of all small dimensions.

Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata