Control systems, signal filters (noise attenuation), modeling epidemics, modeling queues, modeling reliability of repairable systems, modeling recurrent events(such as failures), renewal processes, modeling inventory plans, probability in general (because of the connection with moment generating function) ...

I got taught them in a course on linear systems which was a pre-requisite course to control theory.

Lots of electrical circuits, mechanical systems and electro-mechanical systems can be modelled using laplace transforms if they are linear systems.

I did an electrical and electronic engineering degree and we got to skip the tedious differential equation solving lectures that the mechanical, civil and chemical engineers had to attend because of Monsieur Laplace.