Is there a comprehensive list of distributions, e.g. gamma, Poisson, Gaussian, and when you should use each somewhere? My internet searching has been fruitless.
It is impossible to come up with a comprehensive list of distributions, simply because the space of probability distributions is an infinite dimensional space.
There are many sources where you can find the popular ones:
Among many, many, many other compilations.
Even more, there is new journal about newly developed distributions:
In these compilations you will find basic features of the distributions, such as whether they are discrete or continuous, symmetric or asymmetric, heavy-tailed or light-tailed. However, the final choice is up to the user, and this is usually based on their experience, intuition, and etcetera. This idea is summarised in the popular saying "Modelling is an art".
As @Glen_b points out, the question is just too broad. I believe these links provide some helpful references. I believe it would help to know what you are particularly interested in, so that someone can point to actually relevant distributions.
Still, I love this table. It covers most common distributions, and provides with the relationships with their resp. conjugate priors. A more comprehensive diagram is this one. Notice that this diagrams are interactive, so that you can click on each element, and you jump to a description of the distribution.
See also the related entries and references.