Direction of the wavevector? According to this website and several other websites I have come across, the wavevector direction is in the direction of wave propagation.
On the other hand, Wikipedia suggests that

The direction in which the wave vector points must be distinguished from the "direction of wave propagation". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small wave packet will move, i.e. the direction of the group velocity. For light waves, this is also the direction of the Poynting vector. On the other hand, the wave vector points in the direction of phase velocity. In other words, the wave vector points in the normal direction to the surfaces of constant phase, also called wave fronts.

My guess is that all of the websites saying that it points in the direction of wave propagation are simplifying the matter, because most waves (certainly all the ones I can think of) have their lines of constant phase perpendicular to the direction of wave propagation anyway.
I would be grateful if someone could clarify this, and provide an example of a wave where the wavevctor is not parallel to the direction of wave propagation.
 A: Taking this in order:

My guess is that all of the websites saying that it points in the direction of wave propagation are simplifying the matter,

yes

because most waveshave their lines of constant phase perpendicular to the direction of wave propagation anyway

no, but 

(certainly all the ones I can think of) 

probably.

Most simple examples of EM waves do indeed have a common direction of energy propagation and of phase velocity, but this need not be the case. The simplest example where this property breaks is for anisotropic linear media, i.e. media for which the relationships between the polarization $\mathbf P$, the electric field $\mathbf E$, and the $\mathbf D$ field, are still linear, of the form
$$
\mathbf P=\epsilon_0\chi \mathbf E, \quad\text{and}\quad\mathbf D=\epsilon \mathbf E,
$$
but now the susceptibility $\chi$ and the permittivity $\epsilon$ are matrices (also called a rank-2 tensor in this context), so that, say, the $E_x$ component can induce a polarization along $E_y$:
$$
\mathbf D = \begin{pmatrix}
\epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\
\epsilon_{yx} & \epsilon_{zz} & \epsilon_{yz} \\
\epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} 
\end{pmatrix}\begin{pmatrix} E_x \\ E_y \\ E_z \end{pmatrix}.
$$
When $\epsilon$ is diagonal, the medium is isotropic, but if it has nonzero elements then that fails and you can get noncollinear $\mathbf k$ and $\mathbf S$ vectors. For more details see e.g. these notes or these ones.
