I'm looking for the most suitable class to be a dispatcher for AsyncTasks invoked from my Activities.
I think it could be one of these:
You should definitely use a Service.
The main reason behind this -
Service has it's own documented life-cycle, while Application doesn't.
Application instance, like any of your static variables, can be killed by system almost at any time, you will not receive any callback and can't stop this process. So, any unsaved data (all static variables) will be lost.
On the other hand,
Service can't be killed by system silently, at least
onDestroy() method should be called first. Having such callback, you can save your state to some persistent memory (like SharedPreferences, file, database, etc) and restore that state next time your Application or Service starts.