Is there a list of logics? There are a lots of logics. Some of them are:

*

*Propositional logic

*Predicate logic

*Second order logic

*$n$ order logic

*Fuzzy logic

*Modal logic

*Multivalued logic

*etc

So I`d like to know whether there is a list somewhere in a book or on the internet with it.
Also, are there any efforts to unify or classify all of this mess we created by coming up with such a lot of logics? And what are the kinds of logics that are used in the foundations of Mathematics?
 A: There is this webpage by Peter Suber, but it is 20 years out of date (and I personally find it lacking).  There is also a helpful Wikipedia category that lists some more, but is still incomplete (for instance, it's missing hybrid logic and dynamic epistemic logic).
I think the search term to use is "non-standard logics".  And along with ChrisoLosoph's point, I've gotten a lot of mileage from searching "logic and [insert your favorite field here]."
As for the last part of your question, logics are usually classified by their expressivity and complexity.  There is no "Complexity Zoo" for logic (I wish...) -- instead most of these connections are either folklore or in hard-to-find conference papers.  Here are a few pointers:

*

*One of the first things people want to know about a logic is where it stands in relation to first-order logic. These sorts of results are usually folklore, e.g. modal logics are decidable fragments of first-order logic.

*The relationships between different modal logics are very thoroughly studied.  In addition to the classical Frame Correspondence, people have also explored extensions of modal logic that can still be simulated by 'basic' modal logic (here is a survey).

*Evgeny Zolin has put together a very nice web-navigator for different description logics.

*Hybrid logic has served as a sort of bridge between (afaik) temporal, description, and modal logics (see this paper).

I'd love there was a unified wiki for various logics, but for now I hope this helps!
A: I don't know of any such list, and I don't think there is one.
Alternatively, you can take a look at these Stanford Encyclopedia of Philosophy search results and  this Google search, which is essentially what you're looking for.
