  We study the strong convergence order of the Euler-Maruyama scheme for scalar
stochastic differential equations with additive noise and irregular drift. We
provide a general framework for the error analysis by reducing it to a weighted
quadrature problem for irregular functions of Brownian motion. Assuming
Sobolev-Slobodeckij-type regularity of order $\kappa \in (0,1)$ for the
non-smooth part of the drift, our analysis of the quadrature problem yields the
convergence order $\min\{3/4,(1+\kappa)/2\}-\epsilon$ for the equidistant
Euler-Maruyama scheme (for arbitrarily small $\epsilon>0$). The cut-off of the
convergence order at $3/4$ can be overcome by using a suitable non-equidistant
discretization, which yields the strong convergence order of
$(1+\kappa)/2-\epsilon$ for the corresponding Euler-Maruyama scheme.
