The controllability concepts for linear stochastic differential equations, driven by different kinds of noise processes, can be reduced to the partial controllability concepts for the same systems, driven by correlated white noises. Based on this fact, in this article, we study the conditions of exact and approximate controllability for linear stochastic control systems under various kinds of noise processes, including correlated white noises as well as coloured, wide band and shifted white noises. It is proved that such systems are never exactly controllable while their approximate controllability is equivalent to the approximate controllability of the associated linear deterministic systems at all past time moments.