Let $Y\to M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres. If $m\geq 2$ and $n\geq 3$, we classify all linear connections $A(\Gamma,\Lambda,\Theta):TY\to J^1(TY\to Y)$ in $TY\to Y$ (i.e., classical linear connections on $Y$) depending canonically on a system $(\Gamma,\Lambda,\Theta)$ consisting of a general connection $\Gamma:Y\to J^1Y$ in $Y\to M$, a torsion free classical linear connection $\Lambda:TM\to J^1(TM\to M)$ on $M$ and a linear connection $\Theta:VY\to J^1(VY\to Y)$ in the vertical bundle $VY\to Y$.