  A "shear" is a unipotent translate of a cuspidal geodesic ray in the quotient
of the hyperbolic plane by a non-uniform discrete subgroup of PSL(2,R),
possibly of infinite co-volume. We prove the regularized equidistribution of
shears under large translates with effective (that is, power saving) rates. We
also give applications to weighted second moments of GL(2) automorphic
L-functions, and to counting lattice points on locally affine symmetric spaces.
