  In this paper we prove results on Birkhoff and Oseledets genericity along
certain curves in the space of affine lattices and in moduli spaces of
translation surfaces. We also prove applications of these results to dynamical
billiards, mathematical physics and number theory. In the space of affine
lattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every
point on a curve with some non-degeneracy assumptions is Birkhoff generic for
the geodesic flow. This implies almost everywhere genericity for some curves in
the locus of branched covers of the torus inside the stratum $\mathcal{H}(1,1)$
of translation surfaces. For these curves (and more in general curves which are
well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff
genericity) we also prove that almost every point is Oseledets generic for the
Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin.
As applications, we first consider a class of pseudo-integrable billiards,
billiards in ellipses with barriers, which was recently explored by Dragovic
and Radnovic, and prove that for almost every parameter, the billiard flow is
uniquely ergodic within the region of phase space in which it is trapped. We
then consider any periodic array of Eaton retroreflector lenses, placed on
vertices of a lattice, and prove that in almost every direction light rays are
each confined to a band of finite width. This generalizes a phenomenon recently
discovered by Fraczek and Schmoll which could so far only be proved for random
periodic configurations. Finally, a result on the gap distribution of
fractional parts of the sequence of square roots of positive integers, which
extends previous work by Elkies and McMullen, is also obtained.
