  We compute explicitly the absolute contribution of square-tiled surfaces
having a single horizontal cylinder to the Masur-Veech volume of any ambient
stratum of Abelian differentials. The resulting count is particularly simple
and efficient in the large genus asymptotics. Using the recent results of
Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the
large genus asymptotics of Masur-Veech volumes, we derive that the relative
contribution is asymptotically of the order 1/d, where d is the dimension of
the stratum. Similarly, we evaluate the contribution of one-cylinder
square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the
moduli space of quadratic differentials. We combine this count with our recent
result on equidistribution of one-cylinder square-tiled surfaces translated to
the language of interval exchange transformations to compute empirically
approximate values of the Masur-Veech volumes of strata of quadratic
differentials of all small dimensions.
