When is shear useful? I'd never heard of the shear of a vector field until reading this article.
Shear is the symmetric, tracefree part of the gradient of a vector field.
If you were to decompose the gradient of a vector field into antisymmetric ($\propto$ curl), symmetric tracefree (shear), and tracefull(?) ($\propto$ divergence) parts you'd get that it has components:
$$\frac {\partial A_i}{\partial x^j} = \sigma(A)_{ij} - \frac 12 \epsilon_{ijk}(\nabla \times A)_k + \frac 13 \delta_{ij} (\nabla \cdot A)$$
where $\sigma(A)$ is the shear and is defined as
$$\sigma(A)_{ij} = \frac 12 \left(\dfrac {\partial A_i}{\partial x^j} + \frac {\partial A_j}{\partial x^i}\right) - \frac 13 \delta_{ij} (\nabla \cdot A)$$
Is the shear of a vector field only useful in fluid mechanics?  Does it have any use in pure mathematics -- perhaps analysis or differential geometry?  If so, does anyone have a reference?

In the special case of  two dimensions the shear operator is known as the Cauchy-Riemann operator (or $\bar \partial$ operator, or one of two Wirtinger derivatives), and is denoted 
$\dfrac{\partial}{\partial \bar z}$.
It is certainly useful in complex analysis.
The $n$-dimensional case comes up in the theory of quasiconformal maps where the operator $\sigma$ is known as "the Ahlfors operator", even though the concept certainly predates Ahlfors and goes back at least to Cauchy. Sometimes the name is expanded to be more historically accurate. Other times it's called "the distortion tensor" or the $n$-dimensional Cauchy-Riemann operator. Examples of usage:


*

*$L^\infty$−extremal mappings in AMLE and Teichmüller theory by Luca Capogna, page 4.

*Logarithmic potentials, quasiconformal flows, and Q-curvature by 
Mario Bonk, Juha Heinonen, and Eero Saksman, page 10. 

*Another proof of the Liouville theorem by Zhuomin Liu, page 328.

