When is a flow a shear flow? Let us say I have a velocity profile of a flow:
$$\vec u= \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}$$
Under what circumstances would this be called a shear flow? And can we say anything for an arbitrary number of dimensions?
 A: Compute the gradient of this velocity field:
$$\operatorname{grad}{\bf u}=\nabla_i u_j= \begin{bmatrix}
\partial_x v_{x}& \partial_x v_{y}&\partial_x v_{z}\\
\partial_y v_{x}& \partial_y v_{y}&\partial_y v_{z}\\
\partial_z v_{x}& \partial_z v_{y}&\partial_z v_{z}
\end{bmatrix}$$
This matrix can be represented as a sum of an isotropic, antisymmetric and traceless symmetric part. The isotropic part is proportional to the velocity divergence and is zero for incompressible flow. The antisymmetric part represents rigid rotation of the flow (basically, the curl of the velocity field). The rest is shear (the eigenvectors of the last part of the matrix tell you in which direction it flows inwards (compressive) and in which direction it flows outwards (extensile).
What you see in a standard shear cell (parallel translation of two boundary plates) is actually a combination of rotation and shear.
Explicitly:
$$\operatorname{grad}{\bf u}=(\operatorname{div}{\bf u})I+\tau+\epsilon$$
where
$$\tau_{ij}=\frac12(\nabla_i u_j-\nabla_j u_i)$$
$$\epsilon_{ij}=\frac12(\nabla_i u_j+\nabla_j u_i)-\frac13(\operatorname{div}{\bf u})\delta_{ij}$$
and of course $\operatorname{div}{\bf u}=\sum_i \nabla_i u_i$.
The divergence part is zero for incompressible flow. The rigid rotation tells you how much a cork would rotate if you put it at a certain point in the liquid (it's the tensor representation of the curl vector). The last part is the shear and is the term that is dissipated by viscous forces. It stands in the Navier-Stokes equation:
$$\frac{D{\bf u}}{Dt}=-\nabla p+\eta\nabla \epsilon$$
where $\eta$ is the viscosity. The left hand side is written in the substantive derivative form do reduce clutter.

To answer your question... I'd call it a shear flow when $\epsilon$ is nonzero. However... that's a local definition. A general velocity profile may shear and rotate in general. So... in that respect, you probably want a narrower definition. In comparison with the standard shear flow between parallel plates, you notice another stricter rule: liquid particles stay on parallel surfaces which just move translationally to each other, and usually, we mean the shear flow to be a steady state (no changes of local velocity). The condition that the velocity of a fluid particle stays the same along it path, can be written as $({\bf u}\cdot\nabla){\bf u}=0$. Together, this means $\frac{D{\bf u}}{Dt}=-\nabla p+\eta\nabla \epsilon=0$. You can consider this a more global definition that restricts the entire flow, not just require local shear.
Precise definitions of shear flow may vary, but usually it's pretty obvious what people mean.
