  It is known in \cite{beccari} that the standard explicit Euler-type scheme
(such as the exponential Euler and the linear-implicit Euler schemes) with a
uniform timestep, though computationally efficient, may diverge for the
stochastic Allen--Cahn equation. To overcome the divergence, this paper
proposes and analyzes adaptive time-stepping schemes, which adapt the timestep
at each iteration to control numerical solutions from instability. The
\textit{a priori} estimates in $\mathcal {C}(\mathcal {O})$-norm and
$\dot{H}^{\beta}(\mathcal{O})$-norm of numerical solutions are established
provided the adaptive timestep function is suitably bounded, which plays a key
role in the convergence analysis. We show that the adaptive time-stepping
schemes converge strongly with order $\frac{\beta}{2}$ in time and
$\frac{\beta}{d}$ in space with $d$ ($d=1,2,3$) being the dimension and
$\beta\in(0,2]$. Numerical experiments show that the adaptive time-stepping
schemes are simple to implement and at a lower computational cost than a scheme
with the uniform timestep.
