  We prove some existence, uniqueness and non-existence results of stochastic
strong solutions for a class of stochastic transport equations with a
$q$-integrable (in time), bounded and $\alpha$-H\"{o}lder continuous (in space)
drift coefficient. More precisely, we show that for a Sobolev differentiable
initial condition, there exists a unique stochastic strong solution when
$\alpha>2/q$, while for $\alpha+1<2/q$ with spatial dimension higher than one,
we can choose proper initial data and drift coefficients so that there is no
stochastic strong solutions.
