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 gotten by the classical Picard-Lefschetz formula.

If $F_0$ is non-semistable, I don't know how to compute its monodromy matrix. For example, in Namikawa and Ueno's paper[1], they can compute the Picard-Lefschetz monodromy matrix for each type of singular fiber of genus 2. It's not clear to me how they did that.

[1] Namikawa, Y. and Ueno, K., The complete classification of fibres in pencils of curves of genus two, Manuscripta math., Vol. 9 (1973), 143-186.