Let $f:X\to B$ be a family of curves of genus $g$ over a smooth curve $B$. Let $F_0$ be a singular fiber.
If $F_0$ is a semistable fiber, the monodromy matrix can be get gotten by the classical Picard-Lefschetz's Picard-Lefschetz formula.
If $F_0$ is non-semistable, I don't know how to compute it's its monodromy matrix. For example, in Namikawa and Ueno's paper, they can compute the Picard -Lefschetz Picard-Lefschetz monodromy matrix for each typ type of singular fiber of genus 2. I don's It's not clear to me how they did that.
 Namikawa, Y. and Ueno, K., The complete classification of fibres in pencils of curves of genus two, Manuscripta math., Vol. 9 (1973), 143-186.