ABSTRACT Gaia DR2 has revealed new small-scale and large-scale patterns in the phase-space distribution of stars in the Milky Way. In cylindrical Galactic coordinates $(R,\phi ,z)$, ridge-like structures can be seen in the $(R,V_\phi)$ plane and asymmetric arch-like structures in the $(V_R,V_\phi)$ plane. We show that the ridges are also clearly present when the third dimension of the $(R,V_\phi)$ plane is represented by $\langle z \rangle$, $\langle V_z \rangle$, $\langle V_R \rangle$, $\langle$Fe/H$\rangle$, and $\langle \alpha /{\rm Fe}\rangle$. The maps suggest that stars along the ridges lie preferentially close to the Galactic mid-plane ($|z|\lt 0.2$ kpc), and have metallicity and $\alpha$ elemental abundance similar to that of the Sun. We show that phase mixing of disrupting spiral arms can generate both the ridges and the arches. It also generates discrete groupings in orbital energy – the ridges and arches are simply surfaces of constant energy. We identify eight distinct ridges in the Gaia DR2 data: six of them have constant energy while two have constant angular momentum. Given that the signature is strongest for stars close to the plane, the presence of ridges in $\langle z \rangle$ and $\langle V_z \rangle$ suggests a coupling between planar and vertical directions. We demonstrate, using N-body simulations that such coupling can be generated both in isolated discs and in discs perturbed by an orbiting satellite like the Sagittarius dwarf galaxy.