One can look up in this table that the inverse Laplace transform of $\exp(-c s)/s$ with $c\in\mathbb{R}$ is given by:
$$\frac{1}{2\pi i}\lim_{T\to \infty}\int_{\gamma-i T}^{\gamma+iT}ds\frac{e^{s(t-c)}}{s}=\theta(t-c)$$
where $\gamma=const.>0$ and $\theta(x)$ is the Heaviside step function. Looking at the integral, one might think that one way to arrive at the result could be to close the integration contour in a half-circle to the left in the complex plane, and collect a residue at $s=0$. However, that does not seem to give a step function. Or maybe I am missing some subtlety? What is a correct way to explicitly calculate this inverse Laplace transform?
You close the contour to the left if $t>c$ and to the right if $t<c$, because you should always use the half-plane where the exponential decays. In the second case, there are no poles inside the contour...