If a KeyedService owns objects that depend on the lifetime of its Profile then it needs to ensure that these objects are destroyed before the Profile is destroyed (e.g. content::WebContents need to be be destroyed before their Profile is destroyed). This can be done by overriding KeyedService::Shutdown - this method will be called before the Profile object is destroyed.
Typically, closing the last Browser window associated with a Profile will start tearing down the Profile. This is undesirable if there are content::WebContents associated with that Profile in standalone (i.e. non-Browser) windows that shouldn't go away without an explicit user action to close such a window. In such cases, Profile destruction can be postponed by holding ScopedProfileKeepAlive.
ScopedProfileKeepAlive is a strong reference to a Profile object. It is very similar to ScopedKeepAlive, which is for the browser process.
If an object is not a KeyedService, but still needs to react to destruction of a specific KeyedService, then it can do so using BrowserContextKeyedServiceShutdownNotifierFactory.
For example, extensions::ExtensionURLLoaderFactory is owned by its remote mojo client (i.e. it is not a KeyedService) but it still wants to avoid processing further chrome-extension:// resource requests after Profile destruction. This is done by registering ExtensionURLLoaderFactory::OnBrowserContextDestroyed to be called when extension-related keyed services get destroyed.
If you really need to directly listen for Profile destruction you can use ProfileObserver::OnProfileWillBeDestroyed, but in general if you need this, it‘s a bad sign, as it means that you’re using a feature that is not properly encapsulated in a keyed service.