  We consider a generic and explicit tamed Euler--Maruyama scheme for
multidimensional time-inhomogeneous stochastic differential equations with
multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic,
H\"older continuous and weakly differentiable in the spatial variables while
the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered
by Krylov and R\"ockner (2005). In the discrete scheme, the drift is tamed by
replacing it by an approximation. A strong rate of convergence of the scheme is
provided in terms of the approximation error of the drift in a suitable and
possibly very weak topology. A few examples of approximating drifts are
discussed in detail. The parameters of the approximating drifts can vary and be
fine-tuned to achieve the standard 1/2-strong convergence rate with a
logarithmic factor. The result is then applied to provide numerical solutions
for stochastic transport equations with singular vector fields satisfying the
aforementioned condition.
