I recommend using a custom Ninject Provider to set up your RavenDB DocumentStore. First place this in your code block that registers your Ninject services.
kernel.Bind<IDocumentStore>().ToProvider<RavenDocumentStoreProvider>().InSingletonScope();
Next, add this class that implements the Ninject Provider.
public class RavenDocumentStoreProvider : Provider<IDocumentStore>
{
var store = new DocumentStore { ConnectionName = "RavenDB" };
store.Conventions.IdentityPartsSeparator = "-"; // Nice for using IDs in routing
store.Initialize();
return store;
}
The IDocumentStore needs to be a singleton, but do not make the IDocumentSession a singleton. I recommend that you simply create a new IDocumentSession using OpenSession() on the IDocumentStore instance Ninject gives you whenever you need to interact with RavenDB. IDocumentSession objects are very lightweight, follow the unit-of-work pattern, are not thread-safe, and are meant to be used and quickly disposed where needed.
As others have done, you might also consider implementing a base MVC controller that overrides the OnActionExecuting and OnActionExecuted methods to open a session and save changes, respectively.