Apache's logging framework for .NET. It's powerful, fast, and fairly flexible.
Ranked in these QuestionsQuestion Ranking
Has all the features you'll most likely need, and is quite configurable. Basically what you'd expect from a very popular and widely-used framework.
Pro Modular and Extendable
Has plugins for most of the things you'll need to get done without having to deal with too much unnecessary functionality.
Pro Widely Used
Using log4net is a safe bet - many other open-source projects use it as well and it'll be just another dependency you don't have to worry about. This also means finding support to fix bugs is a lot easier!
Pro Supports Multiple Frameworks
Con Confusing Documentation
Documentation is quite difficult to navigate through unless you know exactly what you're looking for.
Con Dealing with errors can be a pain
Because of the poor documentation and difficulty of configuration you might end up spending a lot of time just trying to fix things that aren't working.