  Under two different scenarios for the light scalar mesons, we investigate the
transition form factors of $B(B_s)$ mesons decay into a scalar meson in the
perturbative QCD approach. In the large recoiling region, the form factors are
dominated by the short-distance dynamics and can be calculated using
perturbation theory. We adopt the dipole parametrization to recast the $q^2$
dependence of the form factors. Since the decay constants defined by the scalar
current are large, our predictions on the $B\to S$ form factors are much larger
than the $B\to P$ transitions, especially in the second scenario. Contributions
from various light-cone distribution amplitudes (LCDAs) are elaborated and we
find that the twist-3 LCDAs provide more than a half contributions to the form
factors. The two terms of the twist-2 LCDAs give destructive contributions in
the first scenario while they give constructive contributions in the second
scenario. With the form factors, we also predict the decay width and branching
ratios of the semileptonic $B\to Sl\bar\nu$ and $B\to Sl^+l^-$ decays. The
branching ratios of $B\to Sl\bar\nu$ channels are found to have the order of
$10^{-4}$ while those of $B\to Sl^+l^-$ have the order of $10^{-7}$. These
predictions can be tested by the future experiments.
