Linear circuit complexity classes The class $\textrm{NC}^i$ is the class functions computable by circuits families of bounded fan-in, $n^{O(1)}$ size and $O(\log^i(n))$ depth. 
The $\textrm{NC}$-hierarchy is the union of those classes. 

Is there any study of the linear-size variant of this hierarchy? That is circuits families of bounded fan-in, polylog depth and linear size?

I know their exists some work with linear-$\textrm{AC}^0$ but nothing else. Remark that at least linear-$\textrm{NC}^1$ is nontrivial since it contains regular languages (and thus some $\textrm{NC}^1$-complete languages).
It follows from work of Valiant [1, 2] that linear-size $\textrm{NC}^1$ can be simulated by $2^{O(n / \log \log n)}$-size circuits of depth three and unbounded fan-in.

For a nice exposition of this result, see section 3 of the survey by Viola [3].

[1] Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In: Mathematical Foundations of Computer Science 1977. MFCS 1977. Lecture Notes in Computer Science, vol 53. Springer, Berlin, Heidelberg.

[2] Leslie G. Valiant. Exponential lower bounds for restricted monotone circuits. In: Proceedings of the fifteenth annual ACM symposium on Theory of computing (STOC '83). ACM, New York, NY, USA, 110-117.

[3] Emanuele Viola. On the power of small-depth computation
. Foundations and Trends in Theoretical Computer Science, vol. 5, num. 1, pp. 1--72, 2009.