  The present paper is devoted to the numerical approximation of an abstract
stochastic nonlinear evolution equation in a separable Hilbert space
{$\mathrm{H}$}. Examples of equations which fall into our framework include the
GOY and Sabra shell models and { a class of nonlinear heat equations.} The
space-time numerical scheme is defined in terms of a Galerkin approximation in
space and a { semi-implicit Euler--Maruyama scheme in time}. {We prove the
convergence in probability of our scheme by means of an estimate of the error
on a localized set of arbitrary large probability.} Our error estimate is shown
to hold in a more regular space $\mathrm{V}_{\beta}\subset \mathrm{H}$ with
$\beta \in [0,\frac14)$ and { that the explicit rate of convergence of our
scheme depends on this parameter $\beta$. }
